Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Verifying the trigonometric identity $\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$ I have the following trigonometric identity
$$\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$$
I've been trying to verify it for almost 20 minutes but coming up wit... | $$
\frac{\cos}{1-\tan x}\cdot\frac{\cos x}{\cos x} = \frac{\cos^2 x}{\cos x-\sin x}
$$
$$
\cos x\cdot\frac{\cos x-\sin x}{\cos x-\sin x} - \frac{\cos^2 x}{\cos x-\sin x} = \frac{-\sin x\cos x}{\cos x-\sin x}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/418476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Distinguishable telephone poles being painted Each of n (distinguishable) telephone poles is painted red, white, blue or yellow. An odd number are painted blue and an even number yellow. In how many ways can this be done?
Can some give me a hint how to approach this problem?
| The exponential generating function (EGF) for the number of red or white poles is
$$1 + z + \frac{1}{2!}z^2 + \frac{1}{3!}z^3 + \dots = e^z$$
The EGF for the number of blue poles is
$$z + \frac{1}{3!}z^3 + \frac{1}{5!}z^5 + \dots = \frac{1}{2}(e^z-e^{-z})$$
The EGF for the number of yellow poles is
$$1 + \frac{1}{2!}z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Upper bound for expression involving logarithms Let $N = 2^p$ for some $p \in \mathbb{N}$. Find the smallest upper bound for $\frac{N}{2}\log\left(\frac{N}{2}\right) + \frac{N}{4}\log\left(\frac{N}{4}\right) + \ldots + 1$
I guess I could first rewrite this to $\frac{2^p}{2}\log\left(\frac{2^p}{2}\right) + \frac{2^p}{4}... | You want
$\begin{align}
\sum_{k=1}^p \frac{n}{2^k}\ln \frac{n}{2^k}
&=\sum_{k=1}^p \frac{n}{2^k}(\ln n- \ln {2^k})\\
&=\sum_{k=1}^p \frac{n}{2^k}(\ln n- k\ln {2})\\
&=n \ln n\sum_{k=1}^p \frac{1}{2^k}
-n\ln 2\sum_{k=1}^p \frac{k}{2^k} \\
\end{align}
$
For not small $p$,
$\sum_{k=1}^p \frac{1}{2^k} \approx 1$.
Since $\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Elementary lower bounds for $n^{1/n}$ I can show that
$n^{1/n} > 1+1/n$
for integer $n \ge 3$
by completely elementary means -
no logs, exponentials,
or calculus.
Are there better bounds that
can be proved in an elementary way?
Here is my proof:
The bound is equivalent to
$n > \frac{(n+1)^n}{n^n}$
or
$\frac{(n+1)^n}{n... | $$\left(1+\frac1n\right)^n=\sum_{k=0}^n\frac{n!}{k!(n-k)!\cdot n^k}$$
After cancelling $n!$ against $(n-k)!$ we are left with $k$ factors $\le n$ in the numerator. Since we have $k$ factors $=n$ in the denominator, we have $\frac{n!}{k!(n-k)!\cdot n^k}\le \frac1{k!}$. This brings us almost to the series $\sum \frac 1{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the values of the constants in the following identity $2x^3+3x^2-14x-5=(ax+b)(x+3)(x+1)+C$ I'm working through identities but I can't figure out how to get further than multiplying out the above to get :
$$2x^3+3x^2-14x-5=2ax^3+3ax^2+3ax+bx^2+3bx+bx+3b+C$$
can someone give me a hint on what to do next?
| The now deleted answer by amWhy used polynomial long division. Since $(x+3)(x+1)= x^2+4x+3$, if we perform the following polynomial long division
we get rightaway the quotient $2x-5=ax+b$, thus $a=2,b=-5$, and the remainder $10=C$.
have you any tips on spotting when to use this ?
Given two polynomials $A(x)$ and $B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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How can $\left({1\over1}-{1\over2}\right)+\left({1\over3}-{1\over4}\right)+\cdots+\left({1\over2n-1}-{1\over2n}\right)+\cdots$ equal $0$?
How can $\left({1\over1}-{1\over2}\right)+\left({1\over3}-{1\over4}\right)+\cdots+\left({1\over2n-1}-{1\over2n}\right)+\cdots$ equal $0$?
Let
$$\begin{align*}x &= \frac{1}{1} + \fr... | Almost everything in your proof works fine until you write "this looks ok if I interprete it as..."
Until then, you are manipulating infinite sums of series with positive terms, these are extended nonnegative real numbers (numbers $x$ such that $0\leqslant x\leqslant+\infty$, if you like) hence adding them and equatin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/420047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Show that $\frac{x}{1-x}\cdot\frac{y}{1-y}\cdot\frac{z}{1-z} \ge 8$.
If $x,y,z$ are positive proper fractions satisfying $x+y+z=2$, prove that $$\dfrac{x}{1-x}\cdot\dfrac{y}{1-y}\cdot\dfrac{z}{1-z}\ge 8$$
Applying $GM \ge HM$, I get $$\left[\dfrac{x}{1-x}\cdot\dfrac{y}{1-y}\cdot\dfrac{z}{1-z}\right]^{1/3}\ge \dfrac{... | We begin by setting $a=1-x, b=1-y, c=1-z$, and noting that $a+b+c=1$. The conditions of the problem imply that $a,b,c\in (0,1)$.
We now need Maclaurin's inequality: $$\frac{a+b+c}{3}\ge \sqrt{\frac{ab+bc+ac}{3}}\ge \sqrt[3]{abc}$$
Ignoring the middle part temporarily, we have $\frac{1}{3}\ge \sqrt[3]{abc}$ or $3\sqrt[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Explanation of this step in a modular arithmetic problem
The multiplicative inverse of $5$ is $7$, when using mod $34$.
$$\begin{align*}
5\cdot x&=3\\[0.1in]
7\cdot 5\cdot x &=7\cdot 3\\[0.1in]
1\cdot x &=7\cdot 3\\[0.1in]
x&=21
\end{align*}$$
I don't understand this part:
$$\begin{align*}
7\cdot 5\cdot x &=7\cdot ... | Recall that $a\equiv b\bmod n$ precisely when $n$ divides the difference $a-b$. Therefore, we have
$$35\equiv 1\bmod 34.$$
It is also true that if $a\equiv b \bmod n$, then $ac\equiv bc\bmod n$ for any $c$. Therefore, whatever $x$ is,
$$35x\equiv x\bmod 34,$$
so that we can go from
$$35x\equiv 21\bmod 34$$
to
$$x\equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving non homogeneous recurrence relation I am having a hard time understanding these questions. I know I need to find the associated homogeneous recurrence relation first, then its characteristic equation. I cant figure out how to find the particular solution to the non homo recurrence relation though.
Ex: $$a_{n}= ... | Forget all this, use generating functions directly. Define $A(z) = \sum_{n \ge 0} a_n z^n$, write:
$$
a_{n + 2} = 4 a_{n + 1} + 4 a_n + 8 (n + 3) \cdot 2^n
$$
Multiply by $z^n$, sum over $n \ge 0$, recognize:
\begin{align}
\sum_{n \ge 0} a_{n + r} z^n
&= \frac{A(z) - a_0 - a_1 z - \ldots - a_{r - 1} z^{r - 1}}{z^r} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Recursion problem help The following are the teachers example problems. The issue is that I don't understand the exact steps they took to go from $f(0)$ to $f(1)$ to $f(2)$ to $f(3)$. What I'm asking here is if someone could be so kind to show me how the answers for $f(0)\ldots f(3)$ are derived for each problem.
Let $... | The first recurrence is $f(n)=(-1)^nf(n-1)+4n$ for $n\ge 1$, with initial value $f(0)=3$. Just plug in successive values of $n$, starting with $n=1$:
$$\begin{align*}
f(1)&=(-1)^1f(0)+4\cdot1=(-1)(3)+4=-3+4=1\\
f(2)&=(-1)^2f(1)+4\cdot2=1\cdot1+8=9\\
f(3)&=(-1)^3f(2)+4\cdot3=(-1)(9)+12=3\\
f(4)&=(-1)^4f(3)+4\cdot4=1\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A problem on Cauchy sequences Let $\langle x_n\rangle $ be a sequence defined recursively by $ 0<a \le x_1 \le x_2\le b $ and $ x_{n+2} =\sqrt{x_nx_{n+1}} $ for each n $ \in \Bbb N $ show that $|x_{n+2} -x_{n+1}| \le \frac {b}{b+a} |x_n -x_{n+1}|$ Deduce that $\langle x_n \rangle$ is Cauchy and find its limit.
Can so... | We can estimate
\begin{eqnarray}
|x_{n+2}-x_{n+1}| & = & |\sqrt{x_{n+1} x_{n+1}}-\sqrt{x_n x_{n+1}}| = \frac{|x_{n+1}(x_{n+1}-x_n)|}{\sqrt{x_{n+1} x_{n+1}} + \sqrt{x_n x_{n+1}}} \\
& = & \frac{1}{1+\sqrt{x_n/x_{n+1}}}|x_{n+1}-x_n| \leq \frac{1}{1+\sqrt{a/b}}|x_{n+1}-x_n| \\
& \leq & \frac{1}{1+a/b/\sqrt{a/b}} |x_{n+1}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
$\sqrt{x + \sqrt{2x -1}} + \sqrt{x- \sqrt{2x-1}} = A $ I am puzzling over the following problem, which involves an equation of the form:
$$\sqrt{x + \sqrt{2x -1}} + \sqrt{x- \sqrt{2x-1}} = A $$
The problem involves finding real values of x corresponding to
A = $ \sqrt{2}$, A = 1, and A = 2, where the roots must be of ... | We look for real solutions. Let $y=2x-1$. Note that $x\ge \frac{1}{2}$. We have $x=\frac{y+1}{2}$. Then
$$x+\sqrt{2x-1}=\frac{y+1}{2}+\sqrt{y}=\frac{1}{2}(1+\sqrt{y})^2.$$
Taking the square root, we find that
$$\sqrt{x+\sqrt{2x-1}}=\frac{1}{\sqrt{2}}(1+\sqrt{y}).$$
Almost similarly, we find that
$$\sqrt{x-\sqrt{2x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Greatest integer $n$ where $n \lt (\sqrt5 +\sqrt7)^6$ I'm really not sure how to do this. I factored out a power of $3$ and squared so that I have $2^3 (6+\sqrt{35})^3 \gt n$ , and I know that if I can prove that $12^3-1 \le (6+\sqrt{35})^3 \lt 12^3$ I am basically done, but I don't know how to do that. Any help is app... | We want a lower bound on
$\sqrt{1-x}$.
From $\sqrt{1-x} = (1-x)^{1/2}
=1-x/2-x^2/8-x^3/16 -5x^4/128 ...
$
I will try
$1-x/2-x^2/4$.
$\begin{align}
(1-x/2-x^2/4)^2
&=1-x-x^2(1/4+1/2)+x^3/4+x^4/16\\
&=1-x-3x^2/4+x^3/4+x^4/16\\
&= 1-x-x^2(3/4-x/4-x^2/16)\\
&< 1-x\\
\end{align}
$
for $0 < x < 1/2$,
so $\sqrt{1-x} > 1-x/2-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Parametrization of $x^2+y^2=z^2$ How can we show that any point on $x^2+y^2=z^2$ can be written in the form (z $\cos(\theta)$, z $\sin(\theta)$, z) for some $\theta$? Here is how I tried to approach it:
$$(z \cos(\theta))^2+(z \sin(\theta))^2)^2=(z)^2$$
$$z^2 \cos^2(\theta)+z^2 \sin^2(\theta)=z^2$$
$$z^2 (\cos^2(\the... | You have shown that if $x=z\cos\theta$ and $y=z\sin\theta$, then $x^2+y^2=z^2$.
The proof could be shortened a bit. Suppose that $x=z\cos\theta$ and $y=z\sin\theta$. Then
$$x^2+y^2=z^2(\cos^2\theta+\sin^2\theta)=(z^2)(1)=z^2.$$
However, that is not what you are being asked to show. You are asked to show that if $x^2+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate a $A_{x\times y}B_{y\times z}$ where $ A$ and $B$ are matrices, $x\neq z$ I know how to evaluate a Cx,y * Dy,x (C rows are equals to D columns), but how do I evaluate a matrix multiplication in which the involved matrices (A and B) have respectively different number of rows and columns?
Here is the "mas... | Your getting what the indices denote mixed up: $C_{x\times y}\times D_{y \times x} = E_{x\times x}$ denotes the product of a matrix $C$ with $x$ rows and $y$ columns, times a matrix $D$ having $y$ rows and $x$ columns. This results in a matrix $E_{x\times x}$ which has $x$ rows and $x$ columns: i.e. $E$ is then a squar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Polynomials Question: Proving $a=b=c$. Question:
Let $P_1(x)=ax^2-bx-c, P_2(x)=bx^2-cx-a \text{ and } P_3=cx^2-ax-b$
, where $a,b,c$ are non zero reals. There exists a real $\alpha$ such
that $P_1(\alpha)=P_2(\alpha)=P_3(\alpha)$. Prove that $a=b=c$.
The questions seems pretty easy for people who know some kind... | You can do this pretty systematically. The given information is equivalent to the statements that $P_1(\alpha) - P_2(\alpha) = 0$ and $P_2(\alpha) - P_3(\alpha) = 0$; in other words that
$$(a-b)\alpha^2 -(b - c)\alpha -(c - a) = 0$$
$$(b-c)\alpha^2 -(c - a)\alpha -(a - b) = 0$$
Next, it is natural to eliminate $\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/429272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
estimate the population numbers $100$ cookies labeled as $1$-$100$ numbers, and a cookie jar. The interviewer randomly selects a number between $1$-$100$ (say he selected $N$), and put cookies number $1$-$N$ to the jar. You randomly pick a cookie from the jar, it is labeled $5$, estimate how many cookies are in the jar... | Let $X \equiv $ cookie you picked. Note that:
$$
Pr(X=5 \mid N=n)=
\begin{cases}
0 & \text{if } n \in \{1,2,3,4\} \\
1/n & \text{if } n \in \{5,6,...,100\}
\end{cases}
$$
Hence, the expected number of cookies given that you picked one that was labelled $5$ is:
$$ \begin{align*}
E[N \mid X=5] &= \sum_{n=1}^{100} n \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/429982",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to evaluate the derivative$\frac{d}{dx}\left(\ln\sqrt{\frac{4+x²}{4-x²}}\right)$? How can I evaluate this derivative?:
$$\frac{d}{dx}\left(\ln\sqrt{\frac{4+x^2}{4-x^2}}\right)$$
Thank you.
| We'll exploit the properties of logarithms, recalling that $$\ln\left(\frac{a}{b}\right)^b = b \ln\left(\frac ab\right) = b (\ln a - \ln b)$$
$$\begin{align} {\bf f(x)} & = \ln\sqrt{\frac{4+x^2}{4-x^2}} \\ \\
&= \ln\left(\frac{4+x^2}{4-x^2}\right)^{1/2}\\ \\
& = \frac 12\ln\left(\frac{4 + x^2}{4 - x^2}\right)\tag{$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Calculating 7^7^7^7^7^7^7 mod 100 What is
$$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$
I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.
| A quick hand calculation gives
$$\begin{align}
7^1 &\equiv 7 \pmod{100} \\
7^2 &\equiv 49 \pmod{100} \\
7^3 &\equiv 43 \pmod{100} \\
7^4 &\equiv 1 \pmod{100}
\end{align}$$
So it reduces to the problem of calculating the value of $7^{7^{7^{7^{7^7}}}} \pmod 4$. And $7^2 \equiv 1 \pmod 4$, so it reduces to the problem of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$ Problem statement:
Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $.
, $n\in \mathbb{N}$
My progress
LHS is e... | First, note that $2\cdot 4 \cdot 6\cdots (2n)=2^n(n!)$. Next, note that if we multiplied $1\cdot 3\cdot 5\cdots (2n-1)$ by $2\cdot 4\cdot 6\cdots (2n)$, that would exactly fill the gaps and produce $(2n)!$. Hence, the denominator of the LHS is $2^nn!$, while the numerator of the LHS is $\frac{(2n)!}{2^nn!}$ Combinin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 2
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divergence of $\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$ i ran into this question and im sitting on it for a long time.
why does this integral diverge:
$$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$$
thank you very much in advance.
yaron.
| Notice the following:
$(x^2 - x - 2) = (x - 2)(x + 1)$
Therefore notice that:
$\frac{2}{x^2 - x - 2} = \frac{2}{(x-2)(x+1)} $
From here we can do a partial fraction decomposition which basically means we want to find:
$A, B$ such that $\frac{A}{x-2} + \frac{B}{x+1} = \frac{2}{x^2 - x - 2}$
This alternatively means that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Trig substitution $\int x^3 \sqrt{1-x^2} dx$ $$\int x^3 \sqrt{1-x^2} dx$$
$x = \sin \theta $
$dx = \cos \theta d \theta$
$$\int \sin^3 \theta d \theta$$
$$\int (1 - \cos^2 \theta) \sin \theta d \theta$$
$u = \cos \theta$
$du = -\sin\theta d \theta$
$$-\int u^2 du$$
$$\frac{-u^3}{3} $$
$$\frac{\cos^3 \theta}{3}$$
Wi... | Let $x=\sin{\theta}$, then $dx = \cos{\theta} \, d\theta$; the integral becomes
$$\int d\theta \, \sin^3{\theta} \, \cos^2{\theta} = \int d\theta \, \sin^3{\theta} -\int d\theta \, \sin^5{\theta} $$
$$\int d\theta \, \sin^3{\theta} = \int d\theta \, \sin{\theta} (1-\cos^2{\theta}) = -\int d(\cos{\theta}) (1-\cos^2{\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/434765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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The golden ratio and a right triangle Assume the square of the hypotenuse of a right triangle is equal to its perimeter and one of its legs is $1$ plus its inradius(the radius inside the circle inscribed inside the triangle.) Find an expression for the hypotenuse $c$ in terms of the golden ratio.
| For any triangle,
$$
2\times\text{area} = \text{inradius}\times\text{perimeter}
$$
For a right triangle,
$$
2\times\text{area} = ab
$$
We are given
$$
\text{inradius}=a-1\quad\text{and}\quad\text{perimeter}=c^2
$$
Therefore,
$$
\begin{align}
\overbrace{(a-1)}^{\text{inradius}}\overbrace{(a+b+c)}^{\text{perimeter}}&=ab\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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solution of difference equation I am trying to solve the following difference equation:
$$-\frac{\epsilon}{h^2}U_{n+1}+\left(\frac{2\epsilon}{h^2}+\frac{1}{h}\right)U_{n}-\left(\frac{\epsilon}{h^2}+\frac{1}{h}\right)U_{n-1}=0,\mbox{ }\mbox{ }\mbox{ }\mbox{ }U_0=1,\mbox{ }U_1=0.$$
I try $U_{n}=Aw^n$ then I get
$$w_{1,2}... | First of all note that, inside the square root, we have
$$
\left(\frac{2\epsilon}{h^2}+\frac{1}{h}\right)^2-4\frac{\epsilon}{h^2}\left(\frac{\epsilon}{h^2}+\frac{1}{h}\right) = \frac{1}{h^2}
$$
So the answers read
$$w_{1,2}=\frac{\left(\frac{2\epsilon}{h^2}+\frac{1}{h}\right)\pm \frac{1}{h}}{2\frac{\epsilon}{h^2}}.
\Lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/436286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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If $a+\sqrt{b}=c+\sqrt{d}$, is it true that $a=c$ and $b=d$?
Assume that $a,b,c,d$ are all positive integers. If $a+\sqrt{b}=c+\sqrt{d}$, is it true that $a=c$ and $b=d$?
I am grading some problems and I don't think this true, but all of a sudden I am doubting myself.
| If $a+\sqrt{b}=c+\sqrt{d}$, then $(a-c)+\sqrt{b}=\sqrt{d}$, so it clearly suffices to study when is it possible to have
$$a+\sqrt{b}=\sqrt{d}.$$
If $b,d$ are arbitrary positive real numbers, then given any $a$ and $b$ such that $a+\sqrt{b}\geq 0$, then there is a $d\geq 0$, namely $d=(a+\sqrt{b})^2$, such that $\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/436437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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Partial fraction integration $\int \frac{dx}{(x-1)^2 (x-2)^2}$ $$\int \frac{dx}{(x-1)^2 (x-2)^2} = \int \frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}+\frac{D}{(x-2)^2}\,dx$$
I use the cover up method to find that B = 1 and so is C. From here I know that the cover up method won't really work and I have to plug in values... | $$
\frac1{(x-1)^2(x-2)^2}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}+\frac{D}{(x-2)^2}\tag{1}
$$
Multiply both sides by $(x-1)^2$ and evaluate at $x=1$: $B=1$
Multiply both sides by $(x-2)^2$ and evaluate at $x=2$: $D=1$
$$
\frac{1-(x-1)^2-(x-2)^2}{(x-1)^2(x-2)^2}=\frac{A}{x-1}+\frac{C}{x-2}\tag{2}
$$
Multiply both ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Differentiate the following w.r.t. $\tan^{-1} \left(\frac{2x}{1-x^2}\right)$ Differentiate : $$ \tan^{-1} \left(\frac {\sqrt {1+x^2}-1}x\right) \quad w.r.t.\quad \tan^{-1} \left(\frac{2x}{1-x^2}\right) $$
| Let $y=\tan^{-1} \left(\dfrac {\sqrt {1+x^2}-1}x\right)$
and let $u= \tan^{-1} \left(\dfrac{2x}{1-x^2}\right)$.
We want to find $dy/du$. Note that:
$$
\dfrac{dy}{dx} = \dfrac{1}{2(1+x^2)}
$$
similarly for $u$, we obtain:
$$
\dfrac{du}{dx} = \dfrac{2}{1+x^2} \iff \dfrac{dx}{du} = \dfrac{1+x^2}{2}
$$
Hence, by Chain Rul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/437341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve a ratio question Studying for the GRE. In the GRE guide, it says that
If the ratio is $2x:5y$, and this equals the ratio $3:4$, what is the ratio of $x:y$?
I tried cross multiplying but I don't get the answer. It says the answer is $15:8$. I get $8:15$. Which step am I missing?
| We are given: $$\dfrac {2x}{5y} = \frac 34$$
$$2x\cdot (4) = 5y \cdot (3)\tag{1}$$ $$ \iff 8x = 15 y\tag{2: cross-multiplied}$$ $$\iff \frac {8x}{y} = 15\tag{divide by y}$$ $$ \iff \frac xy = \frac{15}{8}\tag{divide by 8}$$
It seems as though you went from $(2)$ to $\dfrac {8x}{15y} = 1$, concluding the ratio is $8:15$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/440566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
solving equations by the method of substitution $\dfrac{a}{x}+\dfrac{b}{y}=\dfrac{a}{2}+\dfrac{b}{3},$
$x+1=y$
We have to solve for $x$ and $y$.I have tried to solve for them by finding value of $x$ or $y$ from the second equation and place them in the second.It is obvious that the answers would be $2$ and $3,$but we n... | $\dfrac{a}{x}+\dfrac{b}{y}=\dfrac{a}{2}+\dfrac{b}{3}$...........$(1)$
$x+1=y$............$(2)$
From (2),$x=y-1$. Substituting it into $(1)$,
$\dfrac{a}{y-1}+\dfrac{b}{y}=\dfrac{a}{2}+\dfrac{b}{3}$
$\implies \dfrac{a}{y-1}-\dfrac{a}{2}=\dfrac{b}{3}-\dfrac{b}{y}$
$\implies \dfrac{2a-ay+a}{2(y-1)}=\dfrac{b(y-3)}{3y}$
$\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/440636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is wrong with this Jordan normal form computation? The question I am working on is to compute the Jordan normal form of $$A := \begin{pmatrix} 2 & 1 & 5 \\ 0 & 1 & 3\\ 1 & 0 & 1\end{pmatrix}.$$ The characteristic polynomial and minimal polynomial of $A$ is $x^{2}(x - 4)$. Then the Jordan normal form of $A$ is give... | To calculate the eigenvectors for $\lambda=0$ you want to find a chain. You need to find $\vec{u}_1$ and $\vec{u}_2$ such that $A\vec{u}_2=\vec{u}_1$ and $A\vec{u}_1=0$. Clearly there exists such a solution as zero is an e-value of $A$ hence $A$ is singular. For example,
$\vec{u}_1 = ( -2, -6, 2)$ and $\vec{u}_2 = (-1,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\sin 10^\circ \sin 20^\circ \sin 30^\circ=\sin 10^\circ \sin 10^\circ \sin 100^\circ$? $\sin 10^\circ \sin 20^\circ \sin 30^\circ=\sin 10^\circ \sin 10^\circ \sin 100^\circ$
This is a competition problem which I got from the book "Art of Problem Solving Volume 2". I'm not sure how to solve it because there... | Write $\displaystyle\sin10\sin100=\sin10\sin(90+10)=\sin10\cos10=\frac{1}{2}\sin20=\sin20\sin30$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/442209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$x \sin x=2$ why is my proof that there no solutions wrong? $\frac 12 x \sin x=1$ . Let's look at a right triangle with base $x$ and altitude $\sin x$ . Then our equation is for the area of this triangle. Let the sides of the triangle be $a=x$ , $b=\sqrt {x^2+sin^2 x}$ , and $c= \sin x$ . According to wikipedia, Heron'... | There are solutions on each interval $\left[2k\pi,2k\pi+\frac\pi2\right]$ for positive integer $k$ by the intermediate value theorem because $x\sin(x)$ is $0$ on the left end and $2k\pi+\frac\pi2$ on the right.
Heron's Formula should be
$$
A=\frac{\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}}{4}
$$
Does that cause the same problem?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/442404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Indefinite integral $\int{\frac{dx}{x^2+2}}$ I cannot manage to solve this integral:
$$\int{\frac{dx}{x^2+2}}$$
The problem is the $2$ at denominator, I am trying to decompose it in something like $\int{\frac{dt}{t^2+1}}$:
$$t^2+1 = x^2 +2$$
$$\int{\frac{dt}{2 \cdot \sqrt{t^2-1} \cdot (t^2+1)}}$$
But it's even hard... | I find it much more versatile when encountering a denominator of the form $x^2 + a^2$, rather than only having learned what to do when $a = 1$, I use the fact that : $$\int \dfrac{dx}{x^2 + a^2} = \dfrac 1a\arctan\left(\frac x{a}\right) + C$$
Why? $$\frac{dx}{x^2+a^2} = \frac{dx}{a^2 \left(\frac{x^2}{a^2} + 1\right)} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/442991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
Volume of a wine barrel This is a famous calculus problem and is stated like this
Given a barrel with height $h$, and a small radius of $a$ and
large radius of $b$. Calculate the volume of the barrel
given that the sides are parabolic.
Now I seem to have solved the problem incorrectly because here it seems 2... | Note that in their case $a$, $b$ are radiuses, not diameters. Another important thing to say is that the formula from that external site is not exact!
For simplicity, take $h=1$. Let's define our function on $[-1/2,1/2]$ as $f(x) = 4(a-b)x^2+b$. We write
$$\frac{V}{\pi}= 2\int _{0}^{1/2}(4(a-b)x^2+b)^2 \mathrm dx= 16 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Double integral $\iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y$ Solve the following double integral
\begin{equation}
\iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y
\end{equation}
where $D: \{(x,y)\mid x^2+y^2\leq y \}$.
Some help please? Thank you very much.
| Since
$$
D=\{(x,y)|\ x^2+y^2\le y\}=\left\{(x,y)|\ x^2+(y-\frac12)^2\le \frac14\right\},
$$
we have $y\in [0,1]$ for all $(x,y) \in D$.
Setting
$$
x=r\cos\theta,\ y=r\sin\theta+\frac12,\quad \theta \in [-\pi/2,3\pi/2],\ r \in [0,\frac12],
$$
we get
\begin{eqnarray}
\int_D|x^3y^3|\,dxdy&=&\int_Dy^3|x|^3\,dxdy=\int_{-\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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convert ceil to floor Mathematically, why is this true?
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
Assume $a$ and $b$ are positive integers.
Is this also true if $a$ and $b$ are real numbers?
| If $\frac{a}{b}$ is not an integer and $\frac{a}{b} > n + \frac{1}{b}$, where $n$ is an integer, then it is easy to see that $$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a}{b}+1-\frac{1}{b}\right\rfloor.$$
If $\frac{a}{b}$ is an integer then the relationship
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/448300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 4
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Integration $\int \frac{\sqrt{x^2-4}}{x^4}$ Problem :
Integrate $\int \frac{\sqrt{x^2-4}}{x^4}$
I tried : Let $x^2-4 =t^2 \Rightarrow 2xdx = 2tdt$
$\int \frac{\sqrt{x^2-4}}{x^4} \Rightarrow \frac{t^3 dt}{\sqrt{t^2+4}(t^4-8t+16)}$
But I think this made the integral too complicated... please suggest how to proceed.. Tha... | $\displaystyle\int\frac{\sqrt{x^2-4}}{x^4}dx=\int\frac{\sqrt{1-\frac{4}{x^2}}}{x^3}dx$
Put $\frac1x=z\implies \frac{-1}{x^2}dx=dz$
So it boils to, $-\displaystyle\int z\sqrt{1-4z^2}dz$
Again put $1-4z^2=t\implies -8zdz=dt$
Hence we get, $\frac18\displaystyle\int \sqrt{t}dt=\frac{1}{12}t^{3/2}+C=\frac{1}{12}(1-4z^2)^{3/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$ I'm trying to simplify the following:
$$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$
I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I multiply the numerator by the reciprocal of the denominator.
The pr... | You have the following fraction to simplify:
$$\begin{align} \frac{3}{\sqrt{5}/5} &=\frac{5\times 3}{\sqrt{5}} \\ &=\frac{15}{\sqrt{5}} \\ &=\sqrt{\bigg(\frac{15}{\sqrt{5}}\bigg)^2} \\ &=\sqrt{\frac{15^2}{\sqrt{5}^2}} \\ &=\sqrt{\frac{225}{5}} \\ &=\sqrt{45} \\ &= \sqrt{9\times 5} \\ &= \sqrt{9}\sqrt{5} \\ &= 3\sqrt{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 4
} |
Differentiating $\tan\left(\frac{1}{ x^2 +1}\right)$ Differentiate: $\displaystyle \tan \left(\frac{1}{x^2 +1}\right)$
Do I use the quotient rule for this question? If so how do I start it of?
| Answering this one too now, I'll write the steps straight away
$$\frac{d}{dx}\left(\tan \frac{1}{1+x^2} \right)=
\frac{d}{dy}\left.\left(\tan y \right)\right|_{y=\frac{1}{1+x^2}}
\cdot\frac{d}{dx}\left( \frac{1}{1+x^2} \right)$$
$$=\sec^2y|_{y=\frac{1}{1+x^2}} \cdot \left( - \frac{2x}{1+x^2} \right)$$
$$=\sec^2 \left( ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Power series for $(1+x^3)^{-4}$ I am trying to find the power series for the sum $(1+x^3)^{-4}$ but I am not sure how to find it. Here is some work:
$$(1+x^3)^{-4} = \frac{1}{(1+x^3)^{4}} = \left(\frac{1}{1+x^3}\right)^4 = \left(\left(\frac{1}{1+x}\right)\left(\frac{1}{x^2-x+1}\right)\right)^4$$
I can now use
$$\frac... | Just use the generalized binomial series. For natural $m$:
$$
(1 + u)^{-m}
= \sum_{k \ge 0} \binom{-m}{k} u^k
= \sum_{k \ge 0} (-1)^k \binom{k + m - 1}{m - 1} u^k
$$
and plug in $u = x^3$, $m = 4$:
\begin{align}
(1 + x^3)^{-4}
&= \sum_{k \ge 0} (-1)^k \binom{k + 3}{3} x^{3 k} \\
&= \sum_{k \ge 0} (-1)^k \frac{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Simplifying a radical to solve a problem $$
L = \sqrt{(x+8)^2 + \left(\dfrac{10(x+8)}{x}\right)^2}
$$
$$
L = \sqrt{(x+8)^2 + \dfrac{100(x+8)^2}{x^2}}
$$
$$
L = \sqrt{(x+8)^2\left(1 + \dfrac{100}{x^2}\right)}
$$
$$
L = (x+8)\sqrt{1 + \dfrac{100}{x^2}}
$$
Here am stuck, the answer is
$$
L = \frac{(x+8)}{x}\sqrt{x^2 + 10... | Find the common denominator in the radicand:
$$\begin{align} L = (x+8)\sqrt{1 + \dfrac{100}{x^2}} & = (x+8)\sqrt{\dfrac{x^2 + 100}{x^2}} \\ \\ & = \dfrac{(x+8)}{x}\sqrt{x^2 + 100} \end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/452360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Functional Equation - Am I right? Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$
So here's my solution,
If $x=y=0$,
$2f(0)=2f(0)^2$
$\implies f(0)=0$ or
$f(0)=1$.
Case $1$: $f(0)=0$
If $y=0$,
$x^2+2f(0)=f(x)(f(x)+f(0))$
$$f(x)^2=x^2$$
Now suppose for the sake of ... | In Case 1, we can substitute $y = -x$ into the original equation. We obtain $$x^2+x^2+2f(-x^2)=0(f(x)+f(-x))$$ which means that $$f(-x^2)=-x^2$$ holds for all $x$. This means that $$f(y)=y \text{ holds for all }y\leq 0.\tag{*}$$
Now, let $x>0$ and $y < -x$. Substitute $x$ and $y$ into the original equation. By $(*)$ th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$ without using trigonometry? $$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$
Is there any way to find the answer without using trigonometry, like this?
Hint by Parth Thakkat:
$$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$
$$ = \int \dfrac {\sqrt{x+1}} {x^{1/2}} \cdot \dfrac{dx} {x^{3}}$$
$$... | Hint:
$$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$
$$ = \int \dfrac {\sqrt{x+1}} {x^{1/2}} \cdot \dfrac{dx} {x^{3}}$$
$$ = \int \sqrt{1+\dfrac 1 x} \cdot \dfrac{dx} {x^{3}}$$
Take $t^2 = 1 + \dfrac 1 x$
and note that $ \dfrac 1 x = t^2 - 1$
More Hint:
$t^2 = 1 + \dfrac 1 x$
$\implies 2tdt = -\dfrac 1 {x^2}dx$
$\implies 2t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Partial Fractions I here would like to clear my doubt on the question below:
$$\frac{1}{x(x-1)(x-2)}\;,$$
that is, we want to bring it into the form:
$$\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2}\;,$$
in which the unknown parameters are $A,B$, and $C$. Multiplying these formulas by $x(x − 1)(x − 2)$ turns both into polynom... | $$A=\frac 12,B-1=C=\frac 12$$
these valuse are correct
from the step:
$$A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1$$
put $x=1,x=2,x=0$ you will get right values
even from this equations you also get same values:
$$\left\{\begin{align*}
&A+B+C = 0\\
&3A+2B+C = 0\\
&2A = 1\;.
\end{align*}\right.$$
from 3rd equation $A=\dfrac 12$
afte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$
Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$
I have tried $\sin^3\alpha(3\sin\alpha - 4 \sin^3\alpha) = 3\sin^4\alpha - 4\sin^6\alpha$ and $\cos^3\alpha(4\cos^3\alpha - 3\cos\alpha) = 4\cos^6\alpha - 3\c... | Using $$\cos3A=4\cos^3A-3\cos A,\sin3A=3\sin A-4\sin^3A,$$
$$4(\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha)$$
$$=\sin3\alpha (3\sin\alpha-\sin3\alpha) + \cos3\alpha (\cos3\alpha+3\cos\alpha)$$
$$= \cos^23\alpha-\sin^23\alpha +3(\cos3\alpha\cos\alpha+\sin3\alpha \sin\alpha)$$
$$=\cos6\alpha+3\cos(3\alpha-\alpha)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Maximum and minimum function on area Find maximum and minimum value of function $f(x,y) = 3x+14y$ on $ \left\{ (x,y): 3x^4 + xy + y^4 =6\right\} $.
I will grateful for hints and yours help.
| It is clear that the tangents of the smooth curve $$3x^4+y^4+xy-6=0\,\,\,\, (1)$$ at the extremal points have to be parallel to the level lines of the target function, i.e. $3x+14y=C$. Using the implicit differentiation and equating the slopes, we obtain the equation $$-\frac {12x^3+y} {4y^3+x}= - \frac 3 {14} \Leftr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A geometric property of the graph of $y = x^2$ Consider $n\geq 3$ and two points $A, B$ on the graph of $y = x^2$. Now choose points $P_1,...,P_{n-2}$ on this graph such that the area of the convex $n$-gon $AP_1...P_{n-2}B$ is maximum (they do exist). Let $S_n$ be the area of this $n$-gon and let $S$ be the area betwee... | The area between $AB$ and the graph is just
$$\int_a^b a^2(1-\frac{x-a}{b-a})+b^2\frac{x-a}{b-a}-x^2=a^2x-\frac{a^2}{b-a}(\frac12 x^2-ax)+\frac{b^2}{b-a}(\frac 12x^2-ax)-\frac13x^3\bigg|_a^b$$
$$=a^2b-a^3+\frac12(b+a)(a-b)^2-\frac13 b^3+\frac13 a^3$$
$$=(a-b)(-a^2+\frac12(a^2-b^2)+\frac 13(a^2+ab+b^2))$$
$$=\frac 16(b-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/456480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$ Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$.
I started off by putting $\gcd(a+b, a-b) = d$.
This implies that there are two relatively prime integers $x_1, x_2$, such that
$dx_1 = a+b$
$dx_2 = a -b$
A... | If $ax+by=1$ then $$\begin{align}2&=2ax+2by \\&=(a-b)x + (a+b)x + (b-a)y+(b+a)y \\&= (a-b)(x-y)+(a+b)(x+y)\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/457296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Trigonometric functions I wonder how does a WolframAlpha get this relation where input is a LHS and output is RHS:
$$\cos^2(x)\cos(2x) = \tfrac{1}{4}\cos(4x) + \tfrac{1}{2}\cos(2x) + \tfrac{1}{4}$$
| \begin{align*}
\cos^2(x)\cos(2x) &= \frac{1}{2}(1+\cos(2x))\cos(2x)\\
&= \frac{1}{2}\cos(2x) +\frac{1}{2}\cos^2(2x) \\
&= \frac{1}{2}\cos(2x) + \frac{1}{4} + \frac{1}{4}\cos(4x)
\end{align*}
by two applications of the double angle formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/458131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find all the values of $x \in \mathbb R$ from an equation Find all the values of $x \in \mathbb{R}$ from this inequality:
$$\left|\frac3{x^3-8}\right|=\left|\frac1{x-2}\right|$$
This is my work:
$$\frac{\left|\frac3{x^3-8}\right|}{\left|\frac1{x-2}\right|}=1$$
$$\left|\frac{3(x-2)}{x^3-8}\right|=1$$
$$\left|\frac {x-2}... | HINT:
$x^3-8=x^3-2^3=(x-2)\{x^2+2x+2^2\}=(x-2)\{(x+1)^2+3\}$
So assuming $x-2\ne0,$ we can safely cancel $x-2$ and utilize $|x^2+2x+4|=x^2+2x+4$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/459552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this the correct solution? Determine the coordinates of the vector $U=(4,5,-3)\;\text{of}\; R^3$ with respect to base ${(1,0,0), (0,1,0), (0,0, 1)}$
$$x(1,0,0) + y (0,1,0) + z (0,0,1) = (4,5, -3)$$
$$(x, 0,0) + (0, y, 0) + (0,0,z) = (4,5, -3)$$
$$x +0 +0 = 4 \Longrightarrow x = 4$$
$$0 +0 + y = 5 \Longrightarrow y =... | Yes, note that definition of representation of a vector $\vec v$ with respect to a basis $B=\left\langle {\vec{\beta}_{1},...,\vec{\beta}_{n}} \right\rangle $ of a $n$-dimensional space $V$ is
$$Rep_{B}(\vec v)=\left( {\begin{array}{*{20}{c}}
{c_{1}} \\
{ \vdots } \\
{ c_{n} }
\end{array}} \right)$$
where $c_{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Expected number of people sitting in the right seats. There was a popular interview question from a while back: there are $n$ people getting seated an airplane, and the first person comes in and sits at a random seat. Everyone else who comes in either sits in his seat, or if his seat has been taken, sits in a random un... | Now that I have cleaned my glasses, I'll try again. Thank you, Lord Farin.
Some observations.
The answer must be greater than 1. The probability that the first person, regardless of the number of people, sits in the proper seat is $\frac{1}{n}$, and there are $n$ people, so that expectation is 1. Even if the first pers... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 4
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Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7) $ The question is to simplify $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7)$ without using a calculator .
My friend has given me ... | Consider Heron's formula: the area of a triangle with sides $a, b, \text{and } c$ is
$$
\sqrt{s(s-a)(s-b)(s-c)}
$$
where $s$ is the semi-perimeter $\frac12 (a + b + c)$.
Let $a, b, \text{and } c$ be $\sqrt{5}, \sqrt{6}, \text{and } \sqrt{7}$. Then the area is the square root of your expression divided by $4$. So, wha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 1
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Factorise: $2a^4 + a^2b^2 + ab^3 + b^4$ Factorize : $$2a^4 + a^2b^2 + ab^3 + b^4$$
Here is what I did:
$$a^4+b^4+2a^2b^2+a^4-a^2b^2+ab^3+b^4$$
$$(a^2+b^2)^2+a^2(a^2-b^2)+b^3(a+b)$$
$$(a^2+b^2)^2+a^2(a+b)(a-b)+b^3(a+b)$$
$$(a^2+b^2)^2+(a+b)((a^2(a-b)) +b^3)$$
$$(a^2+b^2)^2+(a+b)(a^3-a^2b+b^3)$$
At this point I don't... | Different Hint
Since your polynomial is homogeneous, this is equivalent to factoring the degree 4 univariate polynomial $2x^4+x^2+x+1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/465136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
How prove this $|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\frac{1}{20n^3}$ Prove that
$$|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\dfrac{1}{20n^3}$$
let $t=\{n\sqrt{2}\}-\{n\sqrt{3}\}$ and $k=[n\sqrt{3}]-[n\sqrt{2}]$
then we have $$t=k-(\sqrt{3}-\sqrt{2})n=k-\sqrt{5-2\sqrt{6}}n\neq 0$$
so
\begin{align*}
t&=\dfrac{(k-(\sqrt{3}-\sqrt{2})n)(... | What I have done is not a full solution, but it may even lead to the solution of a more general problem. Also, using this method I can get only a lower bound. Here it is.
For any positive integer $n$, let $$\sqrt {3} = \frac {a_n} {n} + \varepsilon_n \qquad \text {and} \qquad \sqrt {2} = \frac {b_n} {n} + \delta_n,$$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
System of quadratic equations $x^2 + y = 4$ and $x + y^2 = 10$ How would you solve the following system of equations:
$$
x^2 + y = 4 \\
x + y^2 = 10
$$
Thanks very much!
I tried defining y in terms of x and then inserting in to the second equation:
$$
y = 4 - x^2 \\
x + (4 - x^2)^2 = 10
$$
Expand the second equation:... | To solve general equations of the form $ax^4+bx^3+cx^2+dx+e=0$ requires quartic formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/467229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Triangle with integral side lengths and $\angle A=3\angle B$
$ABC$ is a triangle with integral side lengths. Given that $\angle A=3\angle B$, find the minimum possible perimeter of $ABC$.
I got this problem from an old book (which did not provide even a hint). I can think of some approaches, but all of them result in... | Giving it a try using elementary techniques.
In $\triangle ABC$, let $B=\theta$, $A=3\theta$, $C=\pi-4\theta$. By sine-rule
$$\frac{a}{\sin 3\theta}=\frac{b}{\sin \theta}=\frac{c}{\sin 4\theta}$$
so that $$a=b(3-4\sin^2\theta) \quad , \quad c=b(4\cos \theta \cos 2\theta)$$
On further simplification,
$$a=b(4\cos^2\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Derivative of $\frac {x\cdot\left(1 - 3x\right)}{\sqrt{x-1}}$ Problem. Find the first derivative of $$ \dfrac {x \left( 1 - 3x \right)}{\sqrt{x-1}} $$
Work. Let $u = x-1$ and $y = \dfrac {(u+1)(-3u-2)}{\sqrt{u}} $
Using the chain rule, I got$$\dfrac{(-9x^2-5x+2)}{(2(x-1)^\frac{3}{2})}$$
But the answer is $$\dfrac{(-9x... | If you have some patience to do directly, you will get it.
$$
\frac{d}{dx}\left(\frac{x(1-3x)}{\sqrt{x-1}}\right) =\frac{\sqrt{x-1}(1-6x) - \frac{1}{2\sqrt{x-1}}(x-3x^2)}{x-1} \to \frac{-9x^2 +13x-2}{2(x-1)^\frac{3}{2}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/467592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Sum of a sequence I need guidance for the following question.
Using the fact that $\sum_1^{\infty}\frac{(-1)^{n+1}}{n}=\log2$, $\sum_1^{\infty}\frac{(-1)^{n}}{n(n+1)}$ equals
$1.$ $1-2\log2$
$2.$ $1+2\log2$
$3.$ $(\log2)^2$
$4.$ $-(\log2)^2$
The given sequence gives us $1-\frac12+\frac13-\frac14+\cdots=log2$, but I a... | $$\sum_{n=1}^\infty\frac{(-1)^n}{n(n+1)} = \sum_{n=1}^\infty(-1)^n\left(\frac{1}{n} - \frac{1}{n+1}\right)$$
This is true because:
$$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$$
$$\sum_{n=1}^\infty\frac{(-1)^n}{n(n+1)} = \sum_{n=1}^\infty-\frac{(-1)^{n+1}}{n} - \frac{(-1)^n}{n+1} $$
First term converges to $- \log{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Can someone explain this trigonometric limit without L'Hopital? I can not solve this limit:
$$\lim \limits_{x\to 0}\frac{x^2}{1-\sec(x)}$$
$$\lim \limits_{x\to 0} \frac{x^2}{1-\sec(x)}=\lim \limits_{x\to 0}\frac {x^2}{1-\sec(x)}\cdot{\frac{1+\sec(x)}{1+\sec(x)}}=\lim \limits_{x\to 0}\frac{x^2(1+\sec(x))}{1-\sec^2(x)}=... | Note:
I fixed an error noted by triple_sec .
$\dfrac{x^2}{1-\sec x}
=\dfrac{x^2}{1-1/\cos x}
=\dfrac{ x^2 \cos x}{\cos x-1}
$.
Using
$\cos(2x)
=\cos^2(x)-\sin^2(x)
=1-2\sin^2(x)
$,
$\cos(x)-1
=-2\sin^2(x/2)
$
(I originally had $+$ here instead of $-$)
so
$\dfrac{x^2}{1-\sec x}
=\dfrac{ x^2 \cos x}{\cos x-1}
=\dfrac{ x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Is $(-3)^n + 5^n$ monotone? How can I establish if this sequence is monotone?
If it isn't, is it permanently monotone from a certain n* to infinite?
| To show that a sequence is stictly increasing, we must show that $a_{n+1} > a_n$ for all $n$.
\begin{array}
1(-3)^{n+1}+5^{n+1} &>& (-3)^n + 5^n \\
5^{n+1} - 5^n &>& (-3)^n - (-3)^{n+1} \\
5^n (5-1) &>& (-3)^n (1+3) \\
5^n &>& (-3)^n
\end{array}
When $n$ is even then we have $5^n > 3^n$ which is clearly true. When $n$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The Laurent series of the digamma function at the negative integers To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function.
Specifically,
$$\begin{align} \psi(z + 1) = \frac{1}{z} + \psi(z) &= \psi(1)... | A few terms for starters....
$$
\Psi \left( x \right) = -{x}^{-1}-\gamma+1/6\,{\pi }^{2}x-\zeta
\left( 3 \right) {x}^{2}+{\frac {1}{90}}\,{\pi }^{4}{x}^{3}-\zeta
\left( 5 \right) {x}^{4}+{\frac {1}{945}}\,{\pi }^{6}{x}^{5}+O
\left( {x}^{6} \right)
$$
$$
\Psi \left( x \right) =- \left( x+1 \right) ^{-1}+1-\gamma+ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/469374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Complex Numbers: Finding solutions to $ (z^2-3z+1)^4 = 1 $ I have to find all solutions to
$$ (z^2-3z+1)^4 = 1 $$
What I thought could work was
$$z^2-3z+1= 1^{1/4} $$
Given that the 4 4th-roots of 1 are $1, i, -i, -1$ my idea was to look at each case separately. Starting with $1$ and with $z=a+bi \quad a,b \in R$:
$$z... | Ron Gordon's comment on the first case is exactly the way to go for the real roots of unity: simply factor the expression. For the two complex roots, the quadratic formula should suffice. You may find it worthwhile to convert the complex radicands to polar form by
$$x+iy=re^{i\theta},\,\,\,r=\sqrt{x^2+y^2},\,\,\,\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If n be a positive integer, then prove by binomial theorem that the integral part of $(7+4\sqrt3)^n$ is an odd number. I wanted to know, how can i prove this?
If $n$ be a positive integer, then prove by binomial theorem that the integral part of $(7+4\sqrt3)^n$ is an odd number.
| I'm working with the hypothesis that "integral part" means "floor".
$$\begin{align}
(7+4\sqrt{3})^n + (7 - 4\sqrt{3})^n &= \sum_{k=0}^n \binom{n}{k}7^{n-k}4^k\sqrt{3}^k + \sum_{k=0}^n \binom{n}{k}7^{n-k}(-1)^k4^k\sqrt{3}^k\\
&= 2\sum_{m=0}^{\lfloor n/2\rfloor} \binom{n}{2m}7^{n-2m}4^{2m}3^m
\end{align}$$
is even. $0 < ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity?
$$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$
Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 +... | using half angle identities,
$$\cot(x)=\frac{1}{\tan(x)}=\frac{\sin(2x)}{1-\cos(2x)}=\frac{\left(\frac{1-\cos(4x)}{2}\right)^{\frac{1}{2}}}{1-\left(\frac{1+\cos(4x)}{2}\right)^\frac{1}{2}}$$
with $x=7.5^o=\frac{\pi}{24}$ and $\cos(\frac{4\pi}{24})=\frac{\sqrt{3}}{2}$ we can substitute and simplify by multiplying throug... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/472594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 8,
"answer_id": 7
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Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$.
Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$.
I think the answer should be in terms of 1 integer variable $\in\mathbb Z$ only. I rewrite the equation to $(x+y)^2+(x-y)^2=2^2$, then by the formula of pythagorean triples, $x+y=u^2-v^2,x-y=2uv,2=u^2+v^2$. How can I proceed? Thanks.
| Solutions to the equation $x^2+y^2=2$ with $x,y\in \mathbb{Q}$ can be parametrized by $$\left(x,y\right)=\left(\frac{1+2t-t^{2}}{1+t^{2}},\ \frac{1-2t-t^{2}}{1+t^{2}}\right),$$ where $t\in \mathbb{Q}$.
This follows by rewriting the equation as $$\left(\frac{x+y}{2}\right)^{2}+\left(\frac{x-y}{2}\right)^{2}=1,$$ and usi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/473280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$
Problem : If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$
Solution: Adding $ \cos x +2 \cos y+3 \cos z=0$ and $\sin x+2 \sin y+3 \sin ... | Hint: use complex exponentials.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/479726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Factoring $a^3-b^3$ I recently got interested in mathematics after having slacked through it in highschool. Therefore I picked up the book "Algebra" by I.M. Gelfand and A.Shen
At problem 113, the reader is asked to factor $a^3-b^3.$
The given solution is:
$$a^3-b^3 = a^3 - a^2b + a^2b -ab^2 + ab^2 -b^3 = a^2(a-b) + ab... | Being able to quickly derive the factorization for a difference of cubes (and most other things) prevented me from having to memorize many things in school. I cannot help but share this with you as it was something I must have done a hundred times in a pinch. I will answer your question by delivering a proper full deri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$ Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds:
$$f(x+y)=f(x^{2}+y^{2}).$$
| For any positives $a,b$ we have:
\begin{aligned} f(a) &=f\left(\frac{a+b}{2}+\frac{a-b}{2}\right) \\
&=f\left(\big(\frac{a+b}{2}\big)^2+\big(\frac{a-b}{2}\big)^2\right) \\
&=f\left(\big(\frac{a+b}{2}\big)^2+\big(\frac{b-a}{2}\big)^2\right) \\
&=f\left(\frac{a+b}{2}+\frac{b-a}{2}\right) \\
&=f(b)\end{aligned}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 3
} |
Solving a radical equation $\sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1}$ $$
\sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1}
$$
How many solutions does it have for $x \in \mathbb{R}$?
I squared it once, then rearranges terms to isolate the radical, then squared again.
I got a linear equation, which yielded $x = \frac54$, but when I put... | As $(x+1)-(x-1)=2$ and given that $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}\ \ \ \ (1)$
$$\sqrt{x+1}+\sqrt{x-1}=\frac2{\sqrt{4x-1}}\ \ \ \ (2)$$
On addition, $$2\sqrt{x+1}=\sqrt{4x-1}+\frac2{\sqrt{4x-1}}=\frac{4x+1}{\sqrt{4x-1}}$$
$$\implies 2\sqrt{x+1} \sqrt{4x-1}=4x+1 \ \ \ \ (3)$$
Squaring we get $4x=5\iff x=\frac54$ which... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
} |
Does the derivative of $x^{-1}$ and of $x^3-x$ equal $-\frac{1}{x^{2}}$ and $3x^2-1$? I want to check my answers concerning the derivative of the following functions: $\displaystyle f(x)= \frac{1}{x}$ and of $\displaystyle j(x)= x^3-x$
$$\displaystyle f(x)= \frac{1}{x}$$
$$\begin{align}
f'(x) & = \lim_{h \to 0} \fra... | The title of your post says the derivative of $\frac 1x$ is $\frac{1}{x^2}$ but in your post, you found that the derivative is $-\frac{1}{x^2}$, which is correct.
Also, in the title you wrote that the derivative of $x^3-x$ is $3x^2$, but you found that it is $3x^2-1$
Your calculations are correct, keep up the good work... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/488787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How prove this is an equilateral triangle in $\Delta ABC$,$AB=c,AC=b,BC=a$and such
$$ab^2\cos{A}=bc^2\cos{B}=ca^2\cos{C}$$
show that
$\Delta ABC$ is an equilateral triangle
this problem I have solution,But not nice, and I think this problem have more nice methods,Thank you everyone.
my solution:
$$ab^2\cdot\dfrac{b^... | Dividing throughout by $abc$, we get
$$ \frac{b^2+c^2 - a^2}{c^2} = \frac{a^2 + c^2 - b^2}{a^2} = \frac{a^2+b^2-c^2}{b^2}.$$
Subtracting 1 from each term,
$$ \frac{b^2- a^2}{c^2} = \frac{ c^2 - b^2}{a^2} = \frac{a^2-c^2}{b^2}.$$
Applying Componendo et dividendo, these fractions are equal to
$$ \frac{ b^2-a^2+c^2-b^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find min $P$: $P=\frac{1}{(a+b)(b+c)}+\frac{1}{(c+a)(a+b)}+(c+1)(3+a+b)$ Let $a,b,c\geq 0$ and $a+b+c=1$. Know that never have two numbers both zero. Find min $P$:
$$P=\frac{1}{(a+b)(b+c)}+\frac{1}{(c+a)(a+b)}+(c+1)(3+a+b)$$
| \begin{align}
P=&\frac{1}{(a+b)(b+c)}+\frac{1}{(c+a)(a+b)}+(c+1)(3+a+b)
\\
=&\frac{1}{1-c}\left(\frac{1}{b+c}+\frac{1}{c+a}\right)+(c+1)(4-c)
\\
\ge&\frac{1}{1-c}\frac{4}{(b+c)+(c+a)}+(c+1)(4-c)
\\
=&\frac{4}{1-c^2}+(c+1)(4-c)=:f(c),
\end{align}
the equality holds iff $a=b$.
$$f'(c)=-2c+3+\frac{8c}{(c^2-1)^2}>0,\quad\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$... | Here you go - this person even made your mistake, then someone else corrected it.
Link
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Show that $x(x+1) = y^4+y^3+ay^2+by+c$ has a finite number of positive integral solutions. More precisely,
If $a$, $b$, and $c$
are integers,
show that
there are only a finite number
of positive integers $x$ and $y$
such that
$x(x+1) = y^4+y^3+ay^2+by+c$.
I have a solution,
which I will show in two days
if no better o... | Here is my answer.
Interestingly, it uses no divisibility properties.
We are looking at
$x(x+1) = y^4+y^3+ay^2+by+c$.
I will show that,
if $a$, $b$, and $c$ are integers,
there are at finite number of solutions in
positive integral $x$ and $y$.
This will be done
by finding bounds for $y$
in terms of $a$, $b$, and $c$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove that for each positive integer m, the number $9\cdot 2^m$ can be written as a sum of three squares of positive integers. Prove that for each positive integer $m$, the number $9\cdot2^m$ can be written as a sum of three squares of positive integers.
I think that induction might be used here.!! I could write the f... | We have $$9\cdot 2^1=18=4^2+1^2+1^2$$ and $$9\cdot 2^2=36=4^2+4^2+2^2.$$ Now, if $$9\cdot 2^m=x^2+y^2+z^2,$$ then $$9\cdot 2^{m+2}=4(x^2+y^2+z^2)=(2x)^2+(2y)^2+(2z)^2,$$ and we are done.
(Note the result also holds with $m=0$, as $9=2^2+2^2+1$. And, yes, if needed, the argument can be formalized as an induction that ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Expressing a $3\times 3$ determinant as the product of four factors I am attempting to express the determinant below as a product of four linear factors
$$\begin{vmatrix}
a & bc & b+c\\
b & ca & c+a\\
c & ab & a+b\\
\end{vmatrix}
=
a\begin{vmatrix}
ca & c+a\\
ab & a+b\\
\end{vmatrix}
-
bc\begin{vmatrix}
b & c+a\\
c & ... | $$\begin{vmatrix}
a & bc & b+c\\
b & ca & c+a\\
c & ab & a+b
\end{vmatrix}
=
\begin{vmatrix}
a & bc & b+c\\
b -a & c(a-b)& a-b\\
c -a& b(a-c)& a-c
\end{vmatrix}=(a-b)(c-a)
\begin{vmatrix}
a & bc & b+c\\
-1&c&1\\
1&-b&-1
\end{vmatrix}$$
$$=(a-b)(c-a)\begin{vmatrix}
a& bc& b+c\\
-1&c&-1\\
0&b-c&0
\end{vmatrix}
=(a-b)(c-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/497498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove $3|n(n+1)(n+2)$ by induction I tried proving inductively but I didn't really go anywhere. So I tried:
Let $3|n(n+1)(n+2)$.
Then $3|n^3 + 3n^2 + 2n \Longrightarrow 3|(n(n(n+3)) + 2)$
But then?
| Here's a somewhat tedious (but fairly elementary) way. By the division algorithm, we can write $n=3k+r$ for some integers $k,r$ with $0\le r\le 2,$ whence $$\begin{align}n(n+1)(n+2) &= (3k+r)(3k+r+1)(3k+r+2)\\ &= (3k+r)(9k^2+6kr+9k+r^2+3r+2)\\ &= 27k^3+27k^2r+27k^2+9kr^2+18kr+6k+r^3+3r^2+2r\\ &= 3(9k^3+9k^2r+9k^2+3kr^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/497859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 6
} |
a - b > 0 algebra correction Simple Algebra
$\frac{1}{2}-\frac{1}{5} > 0$
$\frac{1}{2} > \frac{1}{5}$
$5 >2$
Looks correct, but where am I wrong in this,
$\frac{1}{2}-\frac{1}{5} > 0$
$-\frac{1}{5} > -\frac{1}{2}$
$\frac{1}{5} > \frac{1}{2}$
$ 2 > 5$
| $-1>-2$, but does that imply that $1>2$? In general, if $a<b$ then $-a>-b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/499534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Complex Integral with exponential I've been struggling with this:
$$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
| \begin{align*}
\int_0^\infty \frac{e^{-px}}{x^2 + 1}dx &\overset{(1)}{=} \int_0^\infty \int_0^\infty e^{-px} e^{-sx} \sin(s)ds dx \\
&\overset{(2)}{=} \int_0^\infty \int_0^\infty e^{-(p+s)x} \sin (s)dx ds\\
&\overset{(3)}{=} \int_0^\infty \frac{\sin(s)}{(p+s)} ds \\
&\overset{(4)}{=} \text{Ci}(p) \sin (p)+\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/499651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
What is the probability that the student knew the answer to at least one of the two questions? A student takes a true-false examination containing 20
questions. On looking at the examination the student and that he
knows the answer to 10 of the questions which he proceeds to answer
correctly. He then randomly answers t... | The probability of the teacher selecting 2 question, for which the student don't know the answer is: $\frac{10}{20} \times \frac{9}{19}$. There are $\frac{1}{2}$ chance that he'll know the answer so for the both question the probability is:
$$\frac{10}{20} \times \frac{9}{19} \times \frac{1}{4} = \frac{9}{152}$$
The pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/501835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Help with the Integral of $x \arcsin x\,dx$ So I've started the integration using integration by parts, but I'm stuck on the second part of it.
$$ \int_0^\frac{1}{2} x \sin^{-1}x dx $$
*
*Step 1. Set up my variables
$$ u = \arcsin x $$
$$du = \frac{1}{\sqrt{1-x^{2}}} dx $$
$$ dv = x dx $$
... | If you want to continue with step (3), then you can use integration by parts again with $u=x$
$$ \int \frac{x^2}{\sqrt{1-x^2}} dx = -x\sqrt {-{x}^{2}+1}+\int \!\sqrt {1-{x}^{2}}{dx}.$$
For the last integral, you can use the trig. subs $x=\sin(\theta)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/501975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simplest or nicest proof that $1+x \le e^x$ The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which fits into a few lines, is acc... | Repeatedly using $1 + x \le \left(1 + \frac{x}{2} \right)^2$, we have
\begin{align}
1 + x
\le
\left(1 + \frac x 2\right)^2
\le
\left(1 + \frac x 4\right)^4
\le
\left(1 + \frac x 8\right)^8
\le
\dots
\le
\left(1 + \frac x {2^k}\right)^{2^k}.
\end{align}
Taking the limit of $k \rightarrow \infty$ yields
$$
1 + x \le e^x.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "113",
"answer_count": 27,
"answer_id": 11
} |
Limit $ \sqrt{2\sqrt{2\sqrt{2 \cdots}}}$
Find the limit of the sequence $$\left\{\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots\right\}$$
Another way to write this sequence is $$\left\{2^{\frac{1}{2}},\hspace{5 pt} 2^{\frac{1}{2}}2^{\frac{1}{4}},\hspace{5 pt} 2^{\frac{1}{2}}2^{\frac{1}{4}}2^{\frac{1}{8}... | $$\frac 1 2 + \frac 1 4 + \frac 1 8 + \dots = 1 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/505270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
calculate $x^{206}+x^{200}+x^{90}+x^{84}+x^{18}+x^{12}+x^{6}+1$ given $(x+x^{-1})^2 = 3$ If $\left(x+\dfrac 1 x\right)^2=3$ then the value of $$x^{206}+x^{200}+x^{90}+x^{84}+x^{18}+x^{12}+x^{6}+1.$$
I'm trying to solve it like this $$x^2+\dfrac {1}{x^2}=1\text{ and }; x^6+\dfrac {1}{x^6}=-2$$
then solve the expression ... | Since, we have
$x^2=x^4+1 \implies x^4=x^2-1 \implies x^8 = -x^2$.
Then, we get
$x^6 = x^2 .x^4= -1 $
$x^{12} = x^6 . x^6 = 1$
$x^{18} = x^{12}.x^6= (1)(-1)=-1$
$x^{84}= x^{(18).(4)}.x^{12}=(-1)^4 .(1)=1$
$x^{90}=x^{(18)(5)}=(-1)^5=-1$
$x^{200}=x^{(12)(16)}.x^6.x^2=(1)^{16}.(-1).x^2=-x^2$
$x^{206}=x^{200}.x^6=(-x^2).(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Find a closed form of $\sum_{i=1}^n i^3$ I'm trying to compute the general formula for $\sum_{i=1}^ni^3$. My math instructor said that we should do this by starting with a grid of $n^2$ squares like so:
$$
\begin{matrix}
1^2 & 2^2 & 3^2 & ... & (n-2)^2 & (n-1)^2 & n^2 \\
2^2 & 3^2 & 4^2 ... | Notice
$$k^3 = k(k+1)(k+2) - k(3k+2) = k(k+1)(k+2) - 3k(k+1) + k$$
and following identities for positive integer $m$:
$$\sum_{k=1}^n k(k+1)(k+2)\cdots(k+m-1) = \frac{1}{m+1} n(n+1)(n+2)(n+3)\cdots(n+m)$$
One get
$$\begin{align}
\sum_{k=1}^n k^3 = & \frac{1}{4}n(n+1)(n+2)(n+3) - n(n+1)(n+2) + \frac12 n(n+1)\\
= & \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Help checking proof of reverse triangle inequality $|x| - |y| \le |x + y|$? Let $x, y \in \mathbb{R}$. Prove $|x| - |y| \le |x + y|$.
By the the triangle inequality $|x| + |y| \ge |x + y|$, hence
$$
\begin{align}
&|y| \ge |x+y| - |x| \\
&|x+y| \ge |x+y| - |y| \\
\end{align}
$$
Subtracting the first inequality from th... | Sorry, I can't do comments yet.
Your first line after subtracting inequalities is incorrect.
$$|x+y|−|y|≥|x|−|y|$$
if $x=1/2$, $y=-1/2$ we get $-1/2 \ge 0$
We can't subtract inequalities like that, the inequality we are subtracting will be reversed.
e.g. $3>2$ and $3>1$ so $0=3-3 \not> 2-1=1$
but $2=3-1>2-3=-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/507233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq 27$ How can I prove $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq 27$, given that $(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 9$ and $x+y+z=1$.
I've already tried using that: $\frac{1}{x} +\frac{1}{y} +\frac{1}{z}\geq 9$ But I can't seem to manipulate tha... | Cauchy-Schwarz inequality tells us that
\begin{equation}
\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2 \leq 3\times \left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)\qquad (\star)
\end{equation}
But the left-hand term is $\geq 9^2$, so
$$
\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2} \geq \frac{9^2}{3} = 27.
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/507730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How prove this $\frac{1}{2a^2-6a+9}+\frac{1}{2b^2-6b+9}+\frac{1}{2c^2-6c+9}\le\frac{3} {5}\cdots (1)$ let $a,b,c$ are real numbers,and such $a+b+c=3$,show that
$$\dfrac{1}{2a^2-6a+9}+\dfrac{1}{2b^2-6b+9}+\dfrac{1}{2c^2-6c+9}\le\dfrac{3}
{5}\cdots (1)$$
I find sometimes,and I find this same problem:
let $a,b,c$ are real... | Let $f(x) = \dfrac{1}{x^2+(3-x)^2}$. WLOG let $a \le b \le c$.
We note: $f(x) = f(3-x)$ and
if $x < 0$, then $f(x) < f(-x)$ as is obvious from signs or from $f(x)-f(-x) = \dfrac{12x}{4x^4+81}$. Using these, if $a < 0$, we also note
$$f(a)+f(b)+f(c) < f(-a) + f(3-b)+f(3-c)$$
where the new arguments also fulfil th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/509580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Is $\prod \limits_{n=2}^{\infty}(1-\frac{1}{n^2})=1$ Question is to check if $\prod \limits_{n=2}^{\infty}(1-\frac{1}{n^2})=1$
we have $\prod \limits_{n=2}^{\infty}(1-\frac{1}{n^2})=\prod \limits_{n=2}^{\infty}(\frac{n^2-1}{n^2})=\prod \limits_{n=2}^{\infty}\frac{n+1}{n}\frac{n-1}{n}=(\frac{3}{2}.\frac{1}{2})(\frac{4}... | In fact, Euler discovered that
$$\frac{\sin \pi z}{\pi z} = \prod_{n=1}^\infty (1-z^2/n^2)$$
which we can rearrange to
$$\frac{\sin \pi z}{\pi z (1-z^2)} = \prod_{n=2}^\infty (1-z^2/n^2).$$
By comparison of both sides at $z=1$, your product is $1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/513053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Infinite Series $\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}$ How can we prove the following formula?
$$\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}=\frac{-2m+\sqrt{mn}\pi\cot\left(\sqrt{\frac{n}{m}}\pi\right)+n\pi^2\csc^2\left(\sqrt{\frac{n}{m}}\pi\right)}{4mn^2}$$
What is the general method for finding sums of the form $\sum\... | This sum may be evaluated by considering the following contour integral in the complex plane:
$$\oint_C dz \frac{\pi \cot{\pi z}}{(m z^2-n)^2}$$
where $C$ is a rectangular contour that encompasses the poles of the integrand in the complex plane, up to $z=\pm \left ( N +\frac12\right)$, where we consider the limit as $N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What are the possible values of $a$ such that $f(x) = (x + a)(x + 1991) + 1$ has two integer roots? What are the possible values of $a$ such that $f(x) = (x + a)(x + 1991) + 1$ has two integer roots?
$(x + a)(x + 1991) + 1 = x^2 + (1991 + a)x + (1991a + 1)$
This is of the form $ax^2 + bx + c$. Applying the quadratic fo... | You are right, $m^2-4$ cannot be a square for $m>2$. Hint: $m^2 - (m-1)^2 = 2m-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/515398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Determine the value of $a$ and $b$ such that $(ax+3)/(bx+5)$ has $f (-2)= 1$ and $f '(-1) = 1/4$? I tried using the quotient rule but I honestly have no clue how to do this!
Can someone try to help?
$f(x)=\dfrac{ax+3}{bx+5}$.
Determine $a$ and $b$ such that $f(-2)=1$ and $f'(-1)=\frac14$.
| I’m assuming that despite the missing required parentheses, $f(x)$ is supposed to be
$$f(x)=\frac{ax+3}{bx+5}\;.$$
You do indeed want the quotient rule to differentiate $f$:
$$\begin{align*}
f\,'(x)&=\frac{(bx+5)(ax+3)'-(ax+3)(bx+5)'}{(bx+5)^2}\\\\
&=\frac{a(bx+5)-b(ax+3)}{(bx+5)^2}\\\\
&=\frac{abx+5a-abx-3b}{(bx+5)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/517403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Using Exponential Generating Functions on Counting Problems Is it possible to use exponential generating functions to solve problems where repetition is wanted?
For example, if I wanted to solve the following problem which wants distinct possibilities...
How many different 5-letter words can be formed from the letters... | The generating function for the second problem will provide you with all possible 5-card hands. But we are interested in the case when we have at least one cord of each suit. So we need to make some restrictions. We already know the suit of 4 of the cards (because the conditon states they are different) and there are 4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/518020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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Is it true that if $n$ is even then $\sum_{k=1}^{n}(n \bmod k)<\frac{8}{45}n^2$? Let $f(n,k)$ be the least non-negative integer such that $n\equiv f(n,k) \bmod k.$
$f(10,k)(k=1,2,\cdots,10)=0, 0, 1, 2, 0, 4, 3, 2, 1, 0.$
Hence $$\sum_{k=1}^{10}f(10,k)=1+2+4+3+2+1=13.$$
Question: Is it true that if $n$ is even then$$\... | Apply $n\bmod k = n - k \big\lfloor \frac{n}{k} \big\rfloor$ to $$g(n) = \sum_{k=1}^n (n\bmod k)$$
to get $$g(n) = n^2 - \sum_{k=1}^n k \bigg\lfloor \frac{n}{k} \bigg\rfloor.$$
Introduce $$q(n) = \sum_{k=1}^n k \bigg\lfloor \frac{n}{k} \bigg\rfloor$$
and observe that $$q(n+1)-q(n) = (n+1) \bigg\lfloor \frac{n+1}{n+1} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Use the Chinese remainder theorem to find the general solution of $x \equiv a \pmod {2^3}, \; x \equiv b \pmod {3^2}, \; x \equiv c \pmod {11}$ Help! Midterm exam is coming, but i still unable to solve this simple problem using the Chinese remainder theorem.
$$x \equiv a \pmod {2^3}, \quad x \equiv b \pmod {3^2}, \quad... | The general method to find such a number, is to solve for $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$, where the numbers in brackets represent a number modulo $8$, $9$, $11$. Since the number $(1,1,1) = 1$, it is necessary to solve only for two of them.
For $(1,0,0)$, we start off with $99 = (3, 0, 0)$. Since the first multi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/519930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Prove some divisibility results by induction Please hint me, I have two questions:
Prove by induction that:
1)
$$ {13}^n+7^n+19^n=39k,\,\, n\in\mathbb O$$
in which $\mathbb O$ is the set of odd natural numbers.
2)
$$ 5^{2n}+5^n+1=31t,~n\not=3k, $$ $n\in \mathbb N$
| For one, let $\displaystyle f(n): 13^n+7^n+19^n=39k$ holds true for $n=m$ where $m$ is odd positive integer
$\implies\displaystyle13^m+7^m+19^m=39k_1$
$\implies\displaystyle13^m+7^m+19^m=13k_2\implies 7^m+19^m=13k_3\ \ \ \ (1)$
Now for $n=m+2,$
$\displaystyle13^{m+2}+7^{m+2}+19^{m+2}\equiv7^m\cdot49+19^m\cdot361\pmod{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Problem finding in simple algebra It is given,
$$x= \sqrt{3}+\sqrt{2}$$
How to find out the value of $$x^4-\frac{1}{x^4}$$/
The answer is given $40 \sqrt{6}$ but my answer was not in a square-root form
I have done in thsi way:
$$x+ \frac{1}{x}= 2 \sqrt{3}$$
Then,
$$(x^2)^2-\left(\frac{1}{x^2}\right)^2= \left(x^2 + \fra... | Note that $\cfrac 1x=\cfrac 1{\sqrt 3+\sqrt 2}=\cfrac {\sqrt 3-\sqrt 2}{\sqrt 3-\sqrt 2}\cdot\cfrac 1{\sqrt 3+\sqrt 2}=\sqrt 3-\sqrt 2$
So you need to find $(\sqrt 3+\sqrt 2)^4-(\sqrt 3-\sqrt 2)^4$
Now note that the terms which have even powers of $\sqrt 2$ will cancel, and the odd powers will double - so we are left w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/522712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Proving that $n!≤((n+1)/2)^n$ by induction I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this:
$V(1): 1≤1 \text{ true}$
$V(n): n!≤((n+1)/2)^n$
$V(n+1): (n+1)!≤((n+2)/2)^{(n+1)}$
and I've got : $(((n+1)/2)^n)\cdot(n+1)≤((n+2)/2)^{(n+1)}$ $((n+1)^n)n(... | If you really need induction let it be.
Base is $n = 0$: $0! = 1 \le 1 = \left(\frac12\right)^0$.
By induction hypothesis the inequality holds for $n = k$. Let proove it for $n = k+1$.
$$k! \le \left(\frac{k+1}2\right)^k,\\
k!(k+1) \le \left(\frac{k+1}2\right)^k(k+1),$$
$$(k+1)! \le \frac{(k+1)^{k+1}}{2^k}.\tag{*}$$
No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Solving $\frac{5}{t-3}-2=\frac{30}{t^2-9}$ I need help with solving this equation:
$$
\frac{5}{t-3}-2=\frac{30}{t^2-9}
$$
I tried to solve, but I always get false result. The result should be $-\frac{1}{2}$ but I always get $-\frac{1}{2}$ and $3$.
This is how I did it:
$$\begin{align}
\frac{5}{t-3}-2&=\frac{30}{t^2-9}\... | You have cancelled out $t-3$ which is only allowed if $t-3\ne0\iff t\ne3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/523618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.