Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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How to prove this trignometrical Identities? The following two identities comes from my trigonometry module without any sort of proof,
If $A + B + C = \pi $ then,
$$\tan A + \tan B + \tan C = tan A \cdot tan B \cdot tan C$$
and,
$$ \tan \frac{A}{2} \cdot \tan \frac{B}{2} + \tan \frac{B}{2} \cdot \tan \frac{C}{2} + \tan... | If $A+B+C= \pi \Longrightarrow \tan(A+B) = \tan(\pi -C) =-\tan(C)$. So we have $$\tan(A+B)= \frac{\tan(A) + \tan(B)}{1 - \tan(A)\cdot \tan(B)} = -\tan(C) $$ $$\Longrightarrow \tan(A)+\tan(B) = -\tan(C) \cdot \Bigl[1 - \tan(A)\tan(B)\Bigr]$$ from which the first one follows.
And for the second one, we have $\displaystyl... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\int \frac{1}{\sin x\cos x} dx $ Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However... | $\sin(x)\cos(x) = \frac{1}{2} \sin(2x)$.
$I = 2\int \csc(2x)$ let $u = 2x$ then:
$I = \int \csc(u) du = - \log(\cot(2x) + \csc(2x)) + C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/9075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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$n^2 + 3n +5$ is not divisible by $121$ Question:
Show that $n^2 + 3n + 5$ is not divisible by $121$, where $n$ is an integer.
| As $121=11^2,$ we need $11|(n^2+3n+5)$
Let us find $x,y$ such that $x-y=3,x+y=11\implies x=7,y=4$
$$n^2+3n+5=(n+7)(n-4)+33$$
As $33$ is divisible by $11,$ so must be $(n+7)(n-4)$ to make $11|(n^2+3n+5)$
Now $11|(n-4)\iff 11|(n+7)$ as $(n+7)-(n-4)=11$
So in that case, $11^2|(n+7)(n-4),$ but $11^2\not|33$
| {
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$x^y = y^x$ for integers $x$ and $y$ We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
| Say $x^y = y^x$, and $x > y > 0$. Taking logs, $y \log x = x \log y$; rearranging, $(\log x)/x = (\log y)/y$. Let $f(x) = (\log x)/x$; then this is $f(x) = f(y)$.
Now, $f^\prime(x) = (1-\log x)/x^2$, so $f$ is increasing for $x<e$ and decreasing for $x>e$. So if $x^y = y^x$ has a solution, then $x > e > y$. So $y$ mus... | {
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"url": "https://math.stackexchange.com/questions/9505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "92",
"answer_count": 6,
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How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$ How can we prove the following trigonometric identity?
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
| You can find the solution in this page:
*
*http://natto.2ch.net/math/kako/1002/10029/1002903143.html
Translation of the page into English.
$I = \tan (3π/11) +4 \sin (2π/11)$
and $t = 3π/11 $
$11t = 3π$
⇔ $6t = 3π-5t$
⇒ $\sin (6t) = \sin (3π-5t)$ taking sin of both sides
⇔ $2\sin (3t) \cos (3t) = \sin(5t)$ doubl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 6,
"answer_id": 0
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classification up to similarity of complex n-by-n matrices Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.
| It is an Hoffman Kunze exercise problem. It will be for $3\times 3$ matrices $A$, $A^3=I$.
My answer is, the minimal polynomial of $A$ will divide $X^3-1=0$. Now $x^3-1=(x-1)(x-\omega)(x-\omega^2)$ where $\omega^3=1$. So the minimal polynomial can be of the forms
*
*$m=x-a$
*$m=(x-a)(x-b)$
*$m=(x-a)(x-b)(x-c)$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/13640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Exercise 9.2 from Apostol's Mathematical Analysis book. Uniform convergence of product This is a problem (Exercise 9.2) from Apostol's Mathematical Analysis (second edition) which I couldn't solve.
$\bullet$ Define two sequences $\{f_{n}\}$ and $\{g_{n}\}$ as follows:
$f_{n}(x) = x \Bigl(1 + \frac{1}{n}\Bigr)$ for $x ... | Assume that $h_n \to h$ uniformly on the domain $D$ of $\{h_n\}$, where
$$h_n(x) =
\begin{cases}
\frac{x}{n} \biggl(1 + \frac{1}{n} \biggr), & \text{if } x \text{ is irrational} \\
a + \frac{a}{n} + \frac{a}{b} \biggl(1 + \frac{1}{n} \biggr) \biggl(\frac{1}{n} \biggr), & \text{if } x \text{ is rational, say }... | {
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If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$ then $\tan x + \cot x=?$ Hello :)
I hit a problem.
If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$, then how much is $\tan x + \cot x$?
| HINT.
*
*$\tan{x}+\cot{x} = \frac{1}{\sin{x} \cdot \cos{x}}$
*$(\sin{x}+\cos{x})^{2} -1 = 2 \sin{x} \cdot \cos{x}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I find $\int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$? My question is ; How can I solve the following integral question?
$\displaystyle \int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$
Thanks in advance,
| You've had some time to study this, so let's look closer at the two evident approaches: (a) conversion to polar coordinates, (b) integrate directly with a standard hyperbolic substitution.
(a) Conversion to polar coordinates: Let
$$I = \int_{1/\sqrt{2}}^1 \, \int_{\sqrt{1-x^2}}^x \frac{1}{\sqrt{x^2+y^2}} \text{d}y \, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Approximation For Difference Of Two Sides Of A Triangle I have been trying to derive this approximation but have been unsuccessful in doing so. Any help would be greatly appreciated.
| The vectors ${\mathbf{a}}$ and ${\mathbf{c}}$, in terms of their magnitudes and the angles $\theta$ and $\gamma$, are
$$
{\mathbf{a}} = (a \sin \theta, a \cos \theta)
$$
and
$$
{\mathbf{c}} = (c \cos \gamma, c \sin \gamma).
$$
Assuming $c \ll a$, we can expand $b-a$ in powers of the small parameter $\varepsilon = c... | {
"language": "en",
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A geometric look at $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$? Is there a geometric way of looking at the relationship between the positive real numbers $a$, $b$ and $c$ if $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$?
| If you have two poles of length $a$ and $b$, and string wire from the top of one pole to the bottom of the other, and vice versa, the height $c$ at which the wires intersect will be given by half the harmonic mean of the height of the two poles, $\displaystyle \dfrac{1}{c} = \dfrac{1}{a} + \dfrac{1}{b}$, no matter how... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Limit of algebraic function $\ \lim_{x\to\infty} \sqrt[5]{x^5 - 3x^4 + 17} - x$ How to solve this limit?
$$\lim_{x \to \infty}{\sqrt[5]{x^5 - 3x^4 + 17} - x}$$
| If you want to solve ab initio, then the way to go about is the way Arturo suggested.
Note that $a^5 - b^5 = (a-b)(a^4 + ba^3+ b^2a^2 + b^3a + b^4)$ and hence $$(a-b)= \frac{a^5 - b^5}{a^4 + ba^3+ b^2a^2 + b^3a + b^4}$$
Now take $a=\sqrt[5]{x^5-3x^4+17}$ and $b=x$
$\sqrt[5]{x^5-3x^4+17}-x = \frac{(x^5-3x^4+17)-x^5}{\le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Help solving another integral $\int (2x^2+4x-2)^{-\frac{3}{2}} \ dx$ $$\int (2x^2+4x-2)^{-\frac{3}{2}} \ dx$$
Complete the square?
$$\int \frac{1}{(2(x+1)^2-4)^\frac{3}{2}} \ dx$$
Not sure what do do next, or if I should try something else?
Big help if you can show as "step-by-step" possible as you can.
Thanks in advan... | First, factor out a $2$:
$$(2x^2 + 4x - 2) = 2(x^2+2x-1)\quad\text{so}\quad (2x^2+4x-2)^{-3/2} = 2^{-3/2}(x^2+2x-1)^{-3/2}.$$
Do this because you can pull it out of the integral and it will make things easier.
So, up to a constant, this is the same as doing the integral
$$\frac{1}{2^{3/2}}\int\frac{dx}{(x^2+2x-1)^{3/2}... | {
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"timestamp": "2023-03-29T00:00:00",
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Using generating functions to find a formula for the Tower of Hanoi numbers So the Tower of Hanoi numbers are given by the recurrence $h_n=2h_{n-1}+1$ and $h_1=1$. I let my generating function be
$$
g(x)=\sum h_nx^n
$$
Then
$$
g(x)=\sum h_n x^n=\sum (2h_{n-1}+1)x^n=\sum 2h_{n-1}x^n+\sum x^n=2xg(x)+\frac{1}{1-x}.
... | Setting $$g(x) = \sum_{n=0}^{\infty} h_{n+1} x^n,$$ we obtain using $h_n = 2 h_{n-1} +1$ and $h_1 =1$
$$g(x) = \sum_{n=0}^{\infty} h_{n+1} x^n = h_1 + \sum_{n=1}^{\infty} (2 h_{n} +1) x^n = 1 + \sum_{n=0}^{\infty} (2 h_{n+1} +1) x^{n+1} =1 +2 x g(x) + \frac{x}{1-x}.$$
Solving for $g(x)$, we obtain $$g(x) = \frac{1}{(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/24984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding the distance between two triangles, one inside the other I have two right triangles One is a $6$-$8$-$10$ and inside is a $3$-$4$-$5$ and the space between the two triangles is a uniform amount.
I made a really awkward and weird pic of the diagram and I need to solve for $X$
How would I go about solving this? ... | Draw lines connecting the "respective vertices". Add up the areas to solve for $x$.
Area of a right triangle with the non-hypotenuse sides being $a$ and $b$ is $\frac{1}{2}(a \times b)$ while the area of a trapezium with parallel sides being $a$ and $b$ and the height being $h$ is $\frac{1}{2}h(a+b)$
$$\frac{1}{2}x(5+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/26089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Trivial: Rationalize fraction with a third-degree root This is a pretty trivial question. How do you rationalize a function with a denominator that contains a third degree root?
Edit: My expression is $\displaystyle{\frac{1}{\sqrt[3]{2}-1}}$.
| To rationalize when you have a denominator of the form $a^{1/3}-b^{1/3}$ (as you do here, with $a=2$ and $b=1$), use the identity
$x^3-y^3 = (x-y)(x^2+xy+y^2)$.
So
$$2-1 = \left(\sqrt[3]{2}-\sqrt[3]{1}\right)\left(\sqrt[3]{4}+\sqrt[3]{2}+\sqrt[3]{1}\right) = \left(\sqrt[3]{2} - 1\right)\left(\sqrt[3]{4}+\sqrt[3]{2} + 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove $\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor$? everybody, how can I prove that, for natural $m$ and $n$,
$$
\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor \; ?
$$
Thanks a lot.
| The claim is false for real $n$ and $m$. Take $n = \pi$ and $m = -3$ for a counter-example. It isn't even true for integral $n$ and $m$. Take $n = 3$ and $m = -3$ for a counter-example.
Suppose $n$ and $m$ are positive integers. Let $\chi_{\pm,X}(x)$ denote the characteristic function of $X_{\pm}$ defined to be $1$ if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Solving indetermination in limit I'm trying to solve this limit, but I can't get it out of a $\frac{0}{0}$ indetermination:
$$\displaystyle \lim_{x \to 4} \; \frac{x-4}{5-\sqrt{x^2+9}}$$
Maybe there is something I'm missing. Thanks a lot in advanced.
| Assuming a = numerator conjugated and b = denominator conjugated. You can simplify this by doing the following:
$$
\frac{numerator}{denominator} \times \frac{a}{b} \times \frac{b}{a}
$$
Applying this in your case
$$
\lim_ {x \to 4} \frac{x - 4}{5 - \sqrt{x^2 + 9}} =
$$
$$
\lim_ {x \to 4} \frac{x - 4}{5 - \sqrt{x^2 + 9}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37948",
"timestamp": "2023-03-29T00:00:00",
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How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$? Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$
Apart from induction, I tried with Wolfram Alpha to check the validity, but I can't think of an easy (manual) alte... | Note that
$$\begin{align*}\frac{(n+1)\binom{n}{r}}{r+1} &= \frac{(n+1)n!}{(r+1)r!(n-r)!} = \frac{(n+1)!}{(r+1)!(n-r)!}\\ &= \frac{(n+1)!}{(r+1)!((n+1)-(r+1))!} = \binom{n+1}{r+1}.\end{align*}$$
Therefore,
$$\begin{align*}
(n+1)\sum_{r=0}^{n}(-1)^r\frac{\binom{n}{r}}{r+1} &= \sum_{r=0}^{n}(-1)^r\frac{(n+1)\binom{n}{r}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/38623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
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Determinant of Abstract Matrix Given an $n \times n$ matrix $A$, where $x$ is any real number:
$A = \left[
\begin{array}{ c c c c c c c c }
1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 \\
1 & x & x & x & x & x & \cdots & x \\
1 & x & 2x & 2x & 2x & 2x & \cdots & 2x \\
1 & x & 2x & 3x & 3x & 3x & \cdots & 3x ... | I think what you did is almost perfectly correct. In any case, here is another way that uses induction. Let $A_n$ refer to the $n\times n$ matrix. The base case of the $2\times 2$ matrix is $\det A_2 =(x-1)$.
Now, for the inductive step lets look at $A_{n+1}$ and use the linearity of the determinant on the last entr... | {
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"url": "https://math.stackexchange.com/questions/44885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really have no idea why this statement is true. Can someone please explain ... | I will refer to Qiaochu's excellent answer here as proof that if we define
$$f(N):=\sum\limits_{n=0}^N n^2$$
then $f$ is a polynomial of degree $3$.
It is easy to calculate the first few values of this sum. Namely,
$\begin{align}
f(0) &= 0 \\
f(1) &= 1 \\
f(2) &= 5 \\
f(3) &= 14
\end{align}$
I claim that these four poi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/48080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "145",
"answer_count": 32,
"answer_id": 13
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Limit of this series: $\lim_{n\to\infty} \sum^n_{k=0} \frac{k+1}{3^k}$? Given a series, how does one calculate that limit below? I noticed the numerator is an arithmetic progression and the denominator is a geometric progression — if that's of any relevance —, but I still don't know how to solve it.
$$\lim_{n\to\infty}... | For summing $\sum\frac{k}{3^k}$ use the following formula: $$(1-x)^{-2} = 1 + 2x + 3x^{2} + 4x^{3} + \cdots \qquad \Bigl[\because \small (1-x)^{-n} = 1+nx +\frac{n\cdot (n-1)}{2!}\cdot x^{2} + \cdots \Bigr]$$ Multiplying the above equation by $x$ and then putting $x=\frac{1}{3}$ we have $$\frac{1}{3} + \frac{2}{9}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/52150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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$x^4+4$ is composite for $x>1$
$x^4+4$ is composite for $x>1$
I know the Sophie Germain indentity and the get the factorization $$x^4+4 = (x^2+2-2x)(x^2+2+2x)$$
But I am stuck here. I cannot see any general factor here.
| HINT $\ \ x\:(x+2)+2\ \ge\ x\:(x-2)+2\ \ge\ 2\ $ for $\ x\ge 2$
You may find of interest that this is a special case of a class of cyclotomic factorizations due to Aurifeuille, Le Lasseur and Lucas, the so-called Aurifeuillian factorizations of cyclotomic polynomials $\rm\;\Phi_n(x) = C_n(x)^2 - n\ x\ D_n(x)^2\;$. The... | {
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"url": "https://math.stackexchange.com/questions/53767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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The integral $\int_0^8 \sqrt{x^4+4x^2}\,dx$
$\displaystyle \int\nolimits_0^8 \sqrt{x^4+4x^2}\,dx$.
Alright, so I thought I had this figured out. Here's what I did:
*
*I factor out an $x^2$ to get $\sqrt{x^2(x^2+4)}$.
*I let $x = 2\tan(\theta)$, therefore the integrand is $\sqrt{4\tan^2(\theta) (4\tan^2(\theta) +... | There are other ways of doing this integral, but let me try to fix your attempt, which is certainly a fine idea as far as it goes.
The main problem I spot with your development is that you forgot to change the $dx$ when you did the change of variable. (And you should be able to evaluate $\sec(\arctan a)$ as well; we'l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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Finding the smallest positive integer a Can we find the smallest positive integer $a$ such that $1971|50^n+a.23^n$ where n is odd?
Source:Problem Solving Strategies by Arthur Engel
| Since $50^2\equiv23^2\equiv529\pmod{1971}$ and $(529,1971)=1$, we have
$$
\begin{align}
50^{2n+1}+a\cdot23^{2n+1}&\equiv0\pmod{1971}\\
50\cdot529^n+a\cdot23\cdot529^n&\equiv0\pmod{1971}\\
50+a\cdot23&\equiv0\pmod{1971}
\end{align}
$$
Using the Euclid-Wallis Algorithm
$$
\begin{array}{r}
&&85&1&2&3&2\\\hline
1&0&1&-1&3&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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trigonometric system In order to show that $ e^{ix}+e^{iy}+e^{iz}=0 \Longrightarrow e^{2ix}+e^{2iy}+e^{2iz}=0 $, I want to prove that $ \cos x+\cos y+\cos z=0 $ and $ \sin x+\sin y+\sin z=0 \Longrightarrow \cos 2x+\cos 2y+\cos 2z=0$ and $ \sin 2x+\sin 2y+\sin 2z=0 $
$ \cos 2x=2\cos^2 x-1=2(\cos y+\cos z)^2-1 $
$ \cos 2... | You can prove it with pure complex number manipulations itself (and if needed a "pure trigonometric" proof can be read off from that).
Hint:
If $z_1 + z_2 + z_3 = 0$ then $\overline{z_1} + \overline{z_2} + \overline{z_3} = 0$ and if $|z| = 1$, then $\overline{z} = \frac{1}{z}$.
Now try squaring something and use the ab... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$
If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$.
How to solve this one?
(Here's the source of inspiration for the problem.)
| Here's a geometric way of looking at it... the points where $x,y,z > 0$ where $x^2 + y^2 + z^2 = 27$ is the "upper right" 1/8 of the outside of the sphere centered at the origin, of radius $3\sqrt{3}$. Call this surface $S$. If you connect the corners $(3\sqrt{3},0,0)$, $(0,3\sqrt{3},0)$, and $(0,0,3\sqrt{3})$ of $S$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
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Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
| Let P(n) be the given statement. You'll see why in the following step.
$$P(n):1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}$$
Step 1. Let $n = 1$.
Then $\mathrm{LHS} = 1^3 = 1$, $\mathrm{RHS} = \frac{1^2(1+1)^2}{4} = \frac{4}{4} = 1
$.
So LHS = RHS, and this means P(1) is true!
Step 2. Let $P(n)$ be true for $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "67",
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"answer_id": 9
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Simplify with no calculator $\dfrac{(8^3)(-16)^5}{4(-2)^8}$
$\dfrac{8\cdot8\cdot8\cdot-16\cdot-16\cdot-16\cdot-16\cdot-16}{4\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2}$
$\dfrac{8\cdot8\cdot8\cdot-16\cdot-16\cdot-16\cdot-16\cdot-16}{2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2}$
$\dfrac{8\cd... | Hint: Just take everything to powers of 2 using the laws of exponents
$$\frac{8^3(-16)^5}{4(-2)^8}=\frac{-(2^3)^3(2^4)^5}{2^2\cdot2^8}=?$$
| {
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"source": "stackexchange",
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Solving the equation $x + \sqrt{2x+1} = 7$ I can't solve
$x + \sqrt{2x+1} = 7$.
Well, I know the answer is 4, but that is from just reasoning it out. I can't algebraically solve it.
Thus, a step by step is what I really need.
Thanks in advance!!
| $7-x= \sqrt{2x+1}$
$ (7-x)^2 = 2x+1$
$ x^2-14x+49 = 2x+1 $
$x^2-16x+48=0$
$(x-8)^2-64+48=0$
$(x-8)^2=16$
$x-8=\pm 4$
$x=12$ or $x=4$
But $x=12$ does not work in the original equation. So the answer is $x=4$.
(Or the original equation requires $7-x\ge 0$ and so $x\le 7$.)
| {
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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If $X, Y \sim N(0,1)$, find the CDF of $\alpha X + \beta Y$
Possible Duplicate:
Proof that the sum of two Gaussian variables is another Gaussian
Let $X,Y$ be independent normally distributed $N(0,1)$ random variable, and $\alpha,\beta\in \mathbb{R}$. What is the cumulative distribution function of $\alpha X+\beta Y$... | It looks from your comment as if the meaning of your question is different from what I thought at first. My first answer assumed you knew that the sum of independent normals is itself normal.
You have
$$
\exp\left(-\frac12 \left(\frac{x}{\alpha}\right)^2 \right) \exp\left(-\frac12 \left(\frac{z-x}{\beta}\right)^2 \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/65583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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The product rule with square roots I am suppose to find the derivative of $H(u) = (u - \sqrt{u})(u - \sqrt{u})$
I know the formula is the derivative of the second function times the first function plus the derivative of the first function times the second function.
I know that it will be $(1-(1/2) u^{-1/2})(1-(1/2) u ^... | If you are going to use the Product Rule, you have
$$\begin{align*}
H'(u) &= \left(u-\sqrt{u}\right)'\left(u-\sqrt{u}\right) + \left(u-\sqrt{u}\right)\left(u-\sqrt{u}\right)'\\
&= \left(u - u^{1/2}\right)'\left(u-u^{1/2}\right) + \left(u-u^{1/2}\right)\left(u-u^{1/2}\right)'\\
&= \left( 1 - \frac{1}{2}u^{-1/2}\right... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inverse Laplace Transform -s domain How can I find the inverse Laplace transforms of the following function?
$$ G\left(s\right)=\frac{2(s+1)}{s(s^2+s+2)} $$
I solved so far. After that, how do I do?
$$ \frac{1}{s}+\frac{1}{s^2+s+2}+\frac{s}{s^2+s+2}=G\left( s \right)$$
| Your function $$ G(s) = \frac{2(s+1)}{s(s^2 + s + 2)} $$ has the partial fraction decomposition $$ G(s) = \frac{A}{s} + \frac{Bs + C}{(s^2 + s + 2)} $$The way how I will solve this is to use complex analysis. Your original function can be broken down into three distinct linear factors by solving for the zeros of $(s^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Strong Induction: Every natural number $n\geq 8$ can be represented as $n=3k + 5\ell$ Can you please help me and tell, how should I move on?
Can this be proved by induction?
Every natural number $n\geq 8$ can be represented as $n=3k + 5\ell$.
Thank you in advance
| We can avoid an explicit appeal to induction by using the fact that every natural number $n$ has remainder $0$, $1$, or $2$ on division by $3$. Let $n \ge 8$.
If $n$ has remainder $2$ on division by $3$, then $n-8$ is divisible by $3$, say $n-8=3m$. Represent $8$ using $8=3\cdot 1+5\cdot 1$. Then add $m$ $3$'s.
If $n... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof of $\sum_{0 \le k \le t} {t-k \choose r}{k \choose s}={t+1 \choose r+s+1}$? How do I prove that
$$\sum_{0 \le k \le t} {t-k \choose r}{k \choose s}={t+1 \choose r+s+1} \>?$$
I saw this in a book discussing generating functions.
| Suppose we seek to verify that
$$\sum_{0\le k\le t} {t-k\choose r} {k\choose s}
= {t+1\choose r+s+1}.$$
Introduce
$${t-k\choose r} = {t-k\choose t-k-r} =
\frac{1}{2\pi i}
\int_{|z|=\epsilon}
\frac{1}{z^{t-k-r+1}} (1+z)^{t-k} \; dz.$$
This controls the range so we may extend $k$ to infinity,
getting for the sum
$$\fra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd? Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$?
$x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ and $1=x^{m}-1$, which has solution $p=3$ given by $x=2... | The following result can be found in most introductions to Number Theory.
Theorem: Let $x$ and $n$ be integers greater than $1$. If $x^n-1$ is prime, then $x=2$ and $n$ is prime.
Proof: Note that $x-1$ divides $x^n-1$, for
$$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \cdots + 1).$$
If $x>2$ and $n>1$, each of the above factors of $... | {
"language": "en",
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A pencil approach to find $\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}$ What is the fastest, paper-pencil method of finding $$\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}?$$
This is actually a quantitative aptitude problem, and hence the solutions sho... | Edit: Now in beautiful high-definition technicolor!
$$\sum \limits_{k=1}^n \sqrt{\color{red}1+\color{Green}{\frac{1}{k^2}}+\color{Blue}{\frac{1}{(k+1)^2}}}$$
$$=\sum_{k=1}^n\sqrt{\frac{\color{red}{k^2(k+1)^2}+\color{Green}{(k+1)^2}+\color{Blue}{k^2}}{k^2(k+1)^2}}$$
$$=\sum_{l=1}^n \sqrt{\frac{k^4+2k^3+3k^2+2k+1}{k^2(k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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General formula to obtain triangular-square numbers I am trying to find a general formula for triangular square numbers. I have calculated some terms of the triangular-square sequence ($TS_n$):
$TS_n=$1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056
1882672131025, 63955431761796, 2172602007770041, 7380451... | Since all triangular numbers are of the form $\frac{n(n+1)}{2}$, we must have the condition cited
$$
m^2=\frac{n(n+1)}{2}\tag{1}
$$
Equation $(1)$ is equivalent to
$$
2 = \frac{(2n+1)^2-1}{(2m)^2}\tag{2}
$$
According to standard continued fraction theory, a rational approximation to $\sqrt{2}$ as good as $(2)$ must... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 2
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Characteristic polynomial of companion matrix I have a matrix in companion form,
$$A=\begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots & 1 & -a_{n-1} \end{pmatrix}$$
where $A \in M_{n}$. I want to prove by induction that the characteristic polynomial i... | As suggested in the comment above, expand along the first row:
$$\mathrm{det}(tI_n-A) = \mathrm{det} \begin{pmatrix} t & 0 & \cdots & 0 & a_0 \\
-1 & t & \cdots & 0 & a_1 \\
\vdots & \ddots & \ddots & \vdots & \vdots \\
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Help deriving an function from (Lagrangian?) properties I'm trying to derive a function f(x) that has the following properties:
$$f_x-\frac{f}{x}=g_x$$
(You might call this the Lagrangian?)
where
$$\begin{align*}
f&=f(x)\\
f_x&=\frac{df}{dx}\\
g_x&=\frac{d}{dx}\left(\frac{f}{x}\right)
\end{align*}$$
By rearranging ... | Your derivation is incorrect (you've got a wrong sign). I'll use $f'$ instead of $f_x$ (too many $x$'s around...)
We have:
$$f' - \frac{f}{x} = \frac{d}{dx}\frac{f}{x}.$$
If $f=0$, we get a solution. So assume that $f$ is not always $0$. Then the above equation
is equivalent to
$$\begin{align*}
f' - \frac{f}{x} &= \fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integrate $\int\frac{1}{x^6} \sqrt{(1-x^2)^3} ~ dx$ How to integrate the following?
$$\int\frac{\sqrt{(1-x^2)^3}}{x^6} \;dx .$$
| Let $x = \sin(\theta)$. We then get $dx = \cos(\theta) d \theta$. Hence, $$I= \int \frac{\cos^3(\theta)}{\sin^6(\theta)} \cos(\theta) d \theta = \int \cot^4(\theta) cosec^2(\theta) d \theta$$
Let $\cot(\theta) = t$, then $-cosec^2(\theta) d \theta = dt$. Hence, $$I = -\int t^4 dt = -\frac{t^5}{5} + c = -\frac{\cot^5(\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many factors of $2010^{2010}$ have last digit $2$?
How many factors of $2010^{2010}$ have last digit $2$?
It is not difficult to solve this using python/mathematica, but I want to know how to (smartly) solve this one with paper and pencil?
| Since $2010=2\cdot3\cdot5\cdot67$, the factors of $2010^{2010}$ are all of the form $F=2^a\cdot3^b\cdot5^c\cdot67^d$ with $0\le a,b,c,d\le2010$. Since we want the last digit of $F$ to be $2$, we must have $a\ge1$ and $c=0$.
*
*The last digits of $2^a$ ($a\ge1$) are $2,4,8,6$, and they repeat
periodically.
*The last... | {
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How to determine $x$ and $y$ intercepts for $y = 4(x - 2)^2(x + 2)^3$ I need help to determine $x$ and $y$ intercepts for
$$
y = 4(x - 2)^2(x + 2)^3
$$
I guess my first question is, do I need to get the equation into
$$
ax^3 + bx^2 + cx + d
$$
form before starting?
| The x-intercepts can be calculated like this:
$$ y = 4(x - 2)^2(x + 2)^3 $$
$$ y = 0$$
$$ a \cdot b \cdot c = 0 \rightarrow a = 0 \vee b=0 \vee c=0 $$
$$ 4 = 0 \vee (x-2)^2 = 0 \vee (x+2)^3 = 0$$
$$ None \vee x = 2 \vee x \ -2$$
The y-intercept can be found by substituting $0$ for $x$ in $4(x - 2)^2(x + 2)^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/80313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
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Which is the "fastest" way to compute $\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $?
I am looking for the "fastest" paper-pencil approach to compute $$\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $$
This is a quantitative aptitude problem and the correct/required answer is $3.75$
In addition, I am also interested... | I would do it like this. Using $x \frac{\mathrm{d}}{\mathrm{d} x}\left( x^k \right) = k x^k$, and $\sum_{k=1}^n x^k = x \frac{x^n-1}{x-1}$. Then
$$\begin{eqnarray}
\sum_{k=1}^n \left( a k +b\right) x^k &=& \left( a x \frac{\mathrm{d}}{\mathrm{d} x} + b\right) \circ \sum_{k=1}^n x^k = \left( a x \frac{\mathrm{d}}{\m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compacting a Matrix Suppose you have a matrix that is zero-valued everywhere except the diagonal.
As an example, take the identity matrix, $I$. For this example, let's say you are using the $4\times 4$ version.
Is there an operation that can produce a $2\times 2$ matrix consisting of only the non-zero values? If not,... | Here is a solution that you can find by solving a linear system. First we will obtain a $4\times 4$ matrix that looks like your target matrix ($I_2$) but padded. We let
$$A = \begin{bmatrix}a_1 & 0 & 0 & 0\\0 & a_2 & 0 & 0\\0 & 0 & a_3 & 0\\0 & 0 & 0 & a_4\end{bmatrix} \quad{}\text{and}\quad X = \begin{bmatrix}
x_{1,... | {
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"timestamp": "2023-03-29T00:00:00",
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I have a problem understanding the proof of Rencontres numbers (Derangements) I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation
$$D_{n,0}=\left[\frac{n!}{e}\right]$$
where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equat... | (This argument is adapted from page 195 of Concrete Mathematics, Second Edition)
We start with the more conventional representation for the Rencontres number (subfactorial):
$$D_{n,0}=!n=n!\sum_{k=0}^n \frac{(-1)^k}{k!}$$
We also know that
$$\frac{n!}{e}=n!\sum_{k=0}^\infty \frac{(-1)^k}{k!}$$
The difference is
$$\begi... | {
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"timestamp": "2023-03-29T00:00:00",
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$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$ $5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$
Work out the values of $\frac{1}{x+y}$
| Remark : This answer is based on WolframAlpha calculations .
As Sp300 pointed out in his comment by inspection we can see that $x=-1$ is a solution. I will try to show that this is the only solution.
So :
$$2^y=\frac{7}{10}-5^x \Rightarrow y= \log_2{\left(\frac{7}{10}-5^x\right)}$$
$$5^y=\frac{7}{10}-2^x \Rightarrow y=... | {
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"timestamp": "2023-03-29T00:00:00",
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Problem based on Range Find $a$ and $b$ such that the inequality $a \le 3 \cos{x} + 5\cos\left(x - \frac{\pi}{6}\right) \le b$ holds good for all x.
| We can take for example $a=-100$ and $b=100$. However, it might be interesting to find sharp bounds $a$ and $b$. That is a standard max/min problem. We will solve the problem without using the calculus.
Note that
$$\cos(x-\pi/6)=\cos x\cos(\pi/6)+\sin x\sin(\pi/6)=(\cos x)(\sqrt{3}/2)+(\sin x)(1/2).$$
Thus
$$3\cos ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determinant of a matrix with diagonal entries $a$ and off-diagonal entries $b$ I have the following $n\times n$ matrix:
$$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots & a\end{bmatrix}$$
where $0 < b < a$.
I am interested in the expression for the determ... | Add row 2 to row 1, add row 3 to row 1,..., add row $n$ to row 1, we get
$$\det(A)=\begin{vmatrix}
a+(n-1)b & a+(n-1)b & a+(n-1)b & \cdots & a+(n-1)b \\
b & a & b &\cdots & b \\
b & b & a &\cdots & b \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
b & b & b & \ldots & a \\
\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
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How to get closed form from generating function? I have this generating function:
$$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{
\frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$
and I know that $\frac {1}{\sqrt {1-4\,z}}$ is the generating function for the sequence $\binom {2n} {n}$, and $\frac {1-\... | With this type of problem Lagrange inversion is the preferred
approach. Suppose we seek to extract coefficients from
$$Q(z) = \frac{1}{2}
\left(\frac{1}{\sqrt{1-4z}}-1\right)
\left(\frac{1-\sqrt{1-4z}}{2z}-1\right).$$
The closed form for the coefficients is
$$[z^n] Q(z) = \frac{1}{2\pi i}
\int_{|z|=\epsilon}
\... | {
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How to prove :If $p$ is prime greater than $3$ and $\gcd(a,24\cdot p)=1$ then $a^{p-1} \equiv 1 \pmod {24\cdot p}$? I want to prove following statement :
If $p$ is a prime number greater than $3$ and $\gcd(a,24\cdot p)=1$ then :
$a^{p-1} \equiv 1 \pmod {24\cdot p}$
Here is my attempt :
The Euler's totient function ca... | From $p$ is prime, $p>3$ and $\gcd(a,24p)=1$, we infer $\gcd(a,p)=1$, $\gcd(a,3)=1$, $\gcd(a,8)=1$, and $\gcd(p,24)=1$.
From $\gcd(a,p)=1$ we have $a^{p-1} \equiv 1\pmod{p}$.
From $\gcd(a,3)=1$ we have $a^2 \equiv 1\pmod{3}$, and so $a^{p-1} \equiv 1\pmod{3}$ as $p-1$ is even.
From $\gcd(a,8)=1$ we have $a^2 \equiv 1\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Parallelogram trigonometry (Sorry for the ambiguous title, couldn't think of a better one)
While leafing through a highschool textbook, I found what looked like an interesting question in trigonometry. My trigonometry skills are borderline 0, but I didn't expect it to be too much of a challenge. Well, I was wrong:
The ... | For part A, try counting the area of the parallelogram in two different ways, as suggested by Jim Belk.
For part B, notice that your diagram has $n$ and $m$ reversed, since $m<n$. In particular, $\alpha$ should be opposite $m$, not $n$. The modified version of your formula for $m$ is then
$$m^2 = a^2+b^2 -2ab\cos\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/88356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Coefficients of $(1+x+\dots+x^n)^3$? Consider the following polynomial:
$$ (1+x+\dots+x^n)^3 $$
The coefficients of the expansion for few values of $n$ ($n=1$ to $5$) are:
$$ 1, 3, 3, 1 $$
$$ 1, 3, 6, 7, 6, 3, 1 $$
$$ 1, 3, 6, 10, 12, 12, 10, 6, 3, 1 $$
$$ 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1 $$
$$ 1, 3, 6, 10,... | If the polynomial were replaced by an infinite sum, the coefficient of $x^i$ would be equal to the number of ways to choose $(a,b,c) \in \{0,1,2,...\}^3$ such that $a+b+c=i$. This is equal to the number of ways to choose two numbers $\le i$, or $\frac{1}{2}(i+1)(i+2)$. The only difference here is that you want to lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Two players alternately flip a coin; what is the probability of winning by getting a head?
Two players, $A$ and $B$, alternately and independently flip a coin and the first player to get a head wins. Assume player $A$ flips first. If the coin is fair, what is the probability that $A$ wins?
So $A$ only flips on odd t... | \mathsrc{A_k}={A losing in his first k tosses and B losing in his first k tosses and A winning in his k+1 toss}
$$\begin{align}
P(\text{A winning}) &= P(\text{A winning in his first toss or } \mathrm{A_1} \text{ or } \mathrm{A_2} \text{ or } \ldots)\\
&= P(\text{A winning in his first toss}) + P(\mathrm{A_1}) + P(\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 2
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Continuity of a function I was trying to do an exercise: proving that $\frac{x^2}{1-x^2}$ is continuous on $(0,1)$. I did it but I want to be sure that it's right, could you tell me if my argument is wrong?
$\frac{x^2}{1-x^2}-\frac{a^2}{1-a^2}=\frac{(x+a)(x-a)}{(1-x^2)(1-a^2)}$, now $x+a\leq 1+a$. $1-x^2=1-x^2+a^2-a^2=... | Here is the definition of continuity in terms of the epsilon-delta definition: $f$ is continuous at $a$ if and only if for any $\epsilon>0$, there exists $\delta>0$ such that if $|x-a|<\delta$, then $|f(x)-f(a)|<\epsilon$.
Now we have $f(x)=\displaystyle\frac{x^2}{1-x^2}$. Then for any $a\in(0,1)$, we have (as you ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ I have this exponential equation that I don't know how to solve:
$3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$
I tried to factor out a term, but it does not help. Also, I noticed that:
$2 \cdot 9^... | The following substitution may have to work:
$$2^x=t; ~~3^x=s$$
Note that the equation simplifies to, $$12 t^2-35st+ 18s^2=0$$
This factorises to $$(3t-2s)(4t-9s)=0$$
Therefore,
$$3\cdot2^x=2 \cdot 3^x ~~\text{or}~~2^{x+2}=3^{x+2}$$ Since, $(2,3)=1$, we have that $\boxed{x=1~~ \text{or}~~-2}$
Edited to add:
You can vie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
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Infinite Series: Fibonacci/ $2^n$ I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)
In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each term after the first two is the sum of the two previous terms. What is the sum to infinit... | Let $\displaystyle S = \sum_{n=1}^{\infty} \frac{F_n}{2^n}.$ Then
$$ S = \sum_{n=1}^{\infty} \frac{F_n}{2^n} = \frac{1}{2} + \frac{1}{4} + \sum_{n=3}^{\infty} \frac{F_n}{2^n} = \frac{3}{4} + \sum_{n=3}^{\infty} \frac{F_{n-1}+F_{n-2} }{2^n}$$
$$ = \frac{3}{4} + \frac{1}{2} \sum_{n=3}^{\infty} \frac{ F_{n-1} }{2^{n-1} }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 0
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Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series. Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$
Computing the Fourier series of $f$ and using Parseval's identity, I have computed $\z... | Let $f(t):=t^3\ \ (-\pi\leq t\leq \pi)$, extended to all of ${\mathbb R}$ periodically with period $2\pi$. The Fourier series of this function is
$$t^3=\sum_{k=1}^\infty {2(-1)^{k-1}(k^2\pi^2-6)\over k^3}\sin(kt)\qquad(-\pi\leq t\leq \pi).$$
We now use Parseval's formula
$$\|f\|^2=\sum_{k=0}^\infty |c_k|^2.$$
But
$$\|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
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If $a+b+c+d=16$, then $(a+\frac{1}{c})^2+(c+\frac{1}{a})^2 + (b+\frac{1}{d})^2 + (d+\frac{1}{b})^2 \geq \frac{289}{4}$
If $a,b,c,d$ are positive integers and $a+b+c+d=16$, prove that
$$\left(a+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2+\left(b+\frac{1}{d}\right)^2+\left(d+\frac{1}{b}\right)^2 \geq \frac{289... | Since $f(x)=x^2+\frac{1}{x^2}$ is a convex function, by Jensen and AM-GM we obtain:
$$\sum_{cyc}\left(a+\frac{1}{c}\right)^2=\sum_{cyc}f(a)+2\sum_{cyc}\frac{a}{c}\geq 4f\left(\frac{a+b+c+d}{4}\right)+8=\frac{289}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/119910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
How can we show that $\pi (x+y) - \pi(y) \le \frac{1}{3} x + C$ using the sieve of eratosthenes?
How do we show that For $x,y \ge 0$ real numbers, there exists a constant C suchthat: $$\pi(x+y)-\pi(y) \le \frac{1}{3}x+C$$ Where $\pi(.)$ denotes thes prime counting function, is true?
the hint is to sieve n with $y<... | You are asking us to show that your inequality holds for any $x, y \ge 0$ and for any constant $C$, which is obviously false. Perhaps this is what you mean:
Show that there exists a constant $C$ such that for any real numbers $x, y \ge 0$, $\pi(x+y)-\pi(y) \le \frac{1}{3}x+C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/120270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
How I can find the value of $abc$ using the given equations? If I have been given the value of
$$\begin{align*}
a+b+c&= 1\\
a^2+b^2+c^2&=9\\
a^3+b^3+c^3 &= 1
\end{align*}$$
Using this I can get the value of
$$ab+bc+ca$$
How i can find the value of $abc$ using the given equations?
I just need a hint.
I have tried ... | You can get a term involving $abc$ by cubing $a+b+c$:
$$\begin{align*}
(a+b+c)^3 &= (a+b)^3 + 3(a+b)^2c + 3(a+b)c^2 + c^3\\
&= a^3+3a^2b+3ab^2 + b^3 + 3a^2c+\color{blue}{6abc} + 3b^2c + 3ac^2 + 3bc^2 + c^3.
\end{align*}$$
Now use the other information you have to try to find the value of $abc$.
For example, you know... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$ Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$
Use proof by induction. I tried for $n=1$ and got $\frac{27}{9}=3$, but if I assume for $n$ and show it for $n+1$, I don't know what... | No induction necessary.
$$10 + 3\cdot 4^n + 5 \equiv 1 + 3\cdot 4^n + 5 \equiv 6 + 3\cdot 4^n \pmod9$$
Since everything including the base is divisible by three, this reduces to
$$6 + 3\cdot 4^n \pmod9 \iff 2 + 4^n \equiv 2 + 1 \equiv 0 \pmod3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/120649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 8
} |
The solution set of the equation $|2x - 3| = - (2x - 3)$ The solution set of the equation $\left | 2x-3 \right | = -(2x-3)$ is
$A)$ {$0$ , $\frac{3}{2}$}
$B)$ The empty set
$C)$ (-$\infty$ , $\frac{3}{2}$]
$D)$ [$\frac{3}{2}$, $\infty$ )
$E)$ All real numbers
The correct answer is $C$
my solution:
$\ 2x-3 = -(2x-3)$... | In your second case you write $2x - 3 < 0$. I don't understand how you get $\implies 0 = 0$.
From $2x - 3 < 0$ you get $2x < 3$ and hence $x < \frac{3}{2}$.
Now you take the union of your two sets of solutions to get $x \leq \frac{3}{2}$, or in other words, $x \in (-\infty , \frac{3}{2}]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$? I got stuck at the computation of the sum
$$
\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}.
$$
I think there is no purely combinatorial proof here since the sum can achieve negative values. Could you give me solution, it seems to involve generating functions.
| Indeed, generating function method works. Let $c_n$ denoe the given sum. Then we have
$$\begin{align*}
\sum_{n=0}^{\infty} c_n y^{2n}
&= \sum_{n=0}^{\infty} \sum_{k=0}^{n} \frac{(-1)^k y^{2n}}{(2n-2k)!k!} \int_{0}^{\infty} x^{2n-k} e^{-x} \; dx \\
&= \int_{0}^{\infty} \sum_{k=0}^{n} \sum_{n=0}^{\infty} \frac{(yx)^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 5
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Find one side of a parallelogram given its width and two opposite corners Given a parallelogram with two vertical sides defined by a known width (distance between two parallel sides) and two opposite points, what is the equation for the length of the either side?
Real-world application:
I need to determine the cut ang... | Let $\Delta y = C_y - A_y$ and $\Delta x = C_x - A_x$. Then using the right triangle that you've drawn, the length of the non-vertical sides of the parallelogram is
$$\sqrt{\Delta x^2 + (\Delta y - F)^2}.$$
If you slide your blue "W" all the way up so that it touches the "F" line you've drawn, then you have a new littl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to Derive a Double Angle Identity. How does one derive the following two identities:
$$\begin{align*}
\cos 2\theta &= 1-2\sin^2\theta\\
\sin 2\theta &= 2\sin\theta\cos\theta
\end{align*}$$
| The first one follows from:
$$\cos 2\theta = \cos(\theta +\theta) = \cos \theta \cos \theta - \sin \theta \sin \theta = \cos^2 \theta - \sin^2 \theta .$$ Now use the fact $\cos^2 \theta + \sin^2 \theta =1.$
The second one follows from
$$ \sin 2\theta = \sin(\theta +\theta) = \sin \theta \cos \theta + \cos \theta \si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that $u_n$ is arithmetic sequence if $\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{1}{u_3u_4}+\cdots+\frac{1}{u_{n-1}u_n}=\frac{n-1}{u_1u_n}$ Let $(u_n)$ be a sequence $u_i\neq0$ and
$$\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{1}{u_3u_4}+\cdots+\frac{1}{u_{n-1}u_n}=\frac{n-1}{u_1u_n}$$ for all $n\geq3$
Prove that the s... | You need two key features of an arithmetic progression:
$$a_n = a_1 +d(n-1)$$
and
$$a_n-a_{n-1}=d$$ (which is a consequence of the previous one).
Thus
$$\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{1}{u_3u_4}+....+\frac{1}{u_{n-1}u_n}=\frac{n-1}{u_1u_n}$$
$$d\left(\frac{1}{u_1u_2}+\frac{1}{u_2u_3}+\frac{1}{u_3u_4}+....+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Writing $1$ in form of $\frac{1}{t_1}+\cdots+\frac{1}{t_n}$
Possible Duplicate:
Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$
Can anyone help me with this problem? It's a little strange:
Let $M$ be a natural number. Prove that we can write $1=\frac{1}{... | Hint. $$\frac{1}{n} +\frac{1}{n} = \frac{1}{n} + \frac{1}{n+1}+\frac{1}{n(n+1)}.$$
For example, say $N=3$. Then we can write:
$$\begin{align*}
1 &= \frac{1}{4}+\frac{1}{4} + \frac{1}{4}+\frac{1}{4}\\
&= \frac{1}{4} + \frac{1}{5}+\frac{1}{20} + \frac{1}{5}+\frac{1}{20}+\frac{1}{5}+\frac{1}{20}\\
&= \frac{1}{4}+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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ordinary differential equations test the exactness of the O.D.E $(4xy+2x^2 y)dx+(2x^3+3y^2)dy=0$ and hence find the potential function which is the general solution.I tried to solve it and I reached ending up failing to get the integrating factor.please help me
| Hint:
$(4xy+2x^2y)dx+(2x^3+3y^2)dy=0$
$(2x^3+3y^2)dy=-((2x^2+4x)y)dx$
$(2x^3+3y^2)\dfrac{dy}{dx}=-(2x^2+4x)y$
Let $u=y^2$ ,
Then $\dfrac{du}{dx}=2y\dfrac{dy}{dx}$
$\therefore\dfrac{(2x^3+3y^2)}{2y}\dfrac{du}{dx}=-(2x^2+4x)y$
$(2x^3+3y^2)\dfrac{du}{dx}=-(4x^2+8x)y^2$
$(2x^3+3u)\dfrac{du}{dx}=-(4x^2+8x)u$
This belongs to... | {
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"url": "https://math.stackexchange.com/questions/131194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Integrating $\int_0^{2\pi}\frac{1}{1+8\cos^2\theta}d\theta$ Can someone please help me integrate $$\int_0^{2\pi}\frac{1}{1+8\cos^2\theta}d\theta$$ the question says, as a hint, use $\cos\theta = \frac{z + z^{-1}}{2}$ with $|z|=1$. I'm not really sure where to start.
| Let $p(x) = (x-x_1) (x-x_2) \cdots (x-x_n).$ An application of the product rule shows this useful identity (for all $x$ not a root of $p$): $$\frac{p'(x) }{p(x) }= \sum_{k=1}^n \frac{1}{x-x_k}.$$
Now since $\displaystyle \cos t = \frac{e^{it} + e^{-it}}{2} $ we have $$ 1+ 8\cos^2 \left( \frac{k\pi}{n} \right) = 5 + 2a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/132436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Sum the series: $ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \cdots \ \text{ad inf}$ How do I sum the following series?
$$ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \sin{3\alpha} + \cdots \ \text{ad inf}$$... | Let $$ S = \frac{1}{2}\cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4}\cdot \sin{2\alpha} + \frac{1\cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\cdot \sin{3\alpha} + \cdots \ \text{ad inf}$$ and $$ C = 1 + \frac{1}{2}\cdot \cos\alpha + \frac{1\cdot 3}{2 \cdot 4}\cdot \cos{2 \alpha} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Definition of derivative $f(x) = \sqrt{3-5x}$ I am not sure how to factor this out
$$f(x) = \sqrt{3-5x}$$
I then make it $f(x) = \frac {\sqrt{3-5(x+h)} - \sqrt{3-5x}}{h}$
I tried to multiply by the first time + the second term from the numerator which I called x and y
$$\frac {x - y}{h} \cdot \frac {x + y}{x+y}$$
which... | $$\frac{\sqrt{3-5(x+h)}-\sqrt{3-5x}}{h}\times\frac{\sqrt{3-5(x+h)}+\sqrt{3-5x}}{\sqrt{3-5(x+h)}+\sqrt{3-5x}} $$
$$=\frac{\big(3\color{Red}-5(x+\color{Red}h)\big)-\big(3-5x\big)}{h\big(\sqrt{3-5(x+h)}\color{Red}+\sqrt{3-5x}\big)}=\frac{\color{Red}-5\color{Red}h}{h\big(\sqrt{3-5(x+h)}\color{Red}+\sqrt{3-5x}\big)}.$$
Afte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/141200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Bivariate recurrence relation Consider the following recurrence relation:
$$A(h,0)=1\\
A(h,h)=c^h\\
A(h,r)=A(h-1,r)+(c-1)\cdot A(h-1,r-1).$$
Obviously, this is just a generalization of A008949, where $c=2$. Since I'm pretty sure we're not the first ones dealing with it -- is there some source where it was already solve... | Interesting double recurrence. Define the generating function $g(x, y) = \sum_{r, s} A(r, s) x^r y^s$, write without subtraction in indices:
$$
A(r + 1, s + 1) = A(r, s + 1) + (c - 1) A(r, s)
$$
Multiply by $x^r y^s$ and recognize the sums:
\begin{align}
\sum_{r, s} A(r + 1, s) x^r y^s
&= \frac{1}{x} \left( g(x, y) -... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Approximation for $\pi$ I just stumbled upon
$$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$
which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I wonder if it has been every used in past times (historically). Note that the above might also be relate... | I have not seen it before. Note that $\pi = \sqrt{a} + a$ where $a = (1+2\,\pi -\sqrt {1+4\,\pi })/2$, and what you're saying is that a rational approximation of $a$ is
$9/5$. In fact, we have a continued fraction
$$ a = 1 + \dfrac{1}{1 + \dfrac{1}{3+ \dfrac{1}{1+\dfrac{1}{1139 + \ldots}}}}$$
and $1+1/(1+1/(3+1/1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 0
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Evaluating $\int_{-20}^{20}\sqrt{2+t^2}\,dt$ I have this integral:
$$\int_{-20}^{20}\sqrt{2+t^2}\,dt$$
I tried solving it many times but without success.
The end result is this:
$$2\left( 10\sqrt{402}+\mathop{\mathrm{arcsinh}}(10\sqrt{2})\right).$$
I can't seem to get this end result. I got a few wrong ones but cant fi... | The typical way to start dealing with an integral that has $\sqrt{a^2+t^2}$ (in your case, $a=\sqrt{2}$) is to use a trigonometric substitution or a hyperbolic substitution.
If we use hyperbolic substitution, we want to use the fact that
$$1 + \sinh^2 z = \cosh^2 z.$$
Set $t=\sqrt{2}\sinh z$. Then
$$\sqrt{2 + t^2} = \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Finding square roots of $\sqrt 3 +3i$ I was reading an example, where it is calculating the square roots of $\sqrt 3 +3i$.
$w=\sqrt 3 +3i=2\sqrt 3\left(\frac{1}{2}+\frac{1}{2}\sqrt3i\right)\\=2\sqrt 3(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3})$
Let $z^2=w \Rightarrow r^2(\cos(2\theta)+i\sin(2\theta))=2\sqrt 3(\cos\frac{\pi}... | I hope these will answer your direct questions:
The first step is to find the modulus-argument form: the modulus of the complex number comes from $\sqrt{x^2 + y^2}$.
They have taken this as a factor, leaving an 'obvious' way to find the argument by knowing the trigonometric exact values. It's perhaps easier to find th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/148871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Equality involving Appell hypergeometric function After some algebra, Wolfram online integrator gave me the following:
$$\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, 2-N; \frac{5}{2}; \frac{t}{2}, \frac{t}{1-a}\right)+C$$
where:
*
*$\o... | I found also the following term-by-term integration solution.
One has:
$$(1-a-t)^{N-2}\ \sqrt{2t-t^2} = \sqrt{2}\ (1-a)^{N-2}\ \sqrt{t}\ \left( 1-\frac{t}{1-a}\right)^{N-2}\ \sqrt{1-\frac{t}{2}}$$
and by the binomial theorem:
$$\begin{split}
\left( 1-\frac{t}{1-a}\right)^{N-2} &= \sum_{k=0}^{N-2} (-1)^k\ \binom{N-2}{k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can we produce another geek clock with a different pair of numbers? So I found this geek clock and I think that it's pretty cool.
I'm just wondering if it is possible to achieve the same but with another number.
So here is the problem:
We want to find a number $n \in \mathbb{Z}$ that will be used exactly $k \in \... | For $n=9$ and $k=9$ here is a solution:
$1=\left(9+\frac{9}{\left(9 \times \frac{9}{\left(9+\left(9+\left(9-99\right)\right)\right)}\right)}\right)$
$2=\frac{9}{\left(9+\frac{9}{\left(9-\frac{9}{\left(9 \times \frac{9}{99}\right)}\right)}\right)}$
$3=\left(9-\left(9+\left(9-\frac{9}{\left(9 \times \frac{9}{\left(9+99\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 10,
"answer_id": 0
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Integral of $\int \sqrt{1-4x^2}$ I know I am messing up something with the substitutions but I am not sure what.
$$\int \sqrt{1-4x^2}$$
$$u = 4x, du = 4 \,dx$$
$$\frac{1}{4}\int \sqrt{1-u^2}$$
$u = \sin \theta$
$$\frac{1}{4}\int \sqrt{1-\sin^2 \theta} = \frac{1}{4}\int \sqrt{ \cos^2 \theta} = \frac{1}{4}\int \cos \the... | Let $2x = u$. So, $dx = \frac{1}{2}du$
$$\frac{1}{2}\int \sqrt{1-u^2} du$$
Replace $u = \sin \theta$, $du = cos \theta\ d\theta$
So, equation will be
$$\frac{1}{2}\int \sqrt{1-sin^2 \theta}\ cos\theta\ d\theta$$
$$\frac{1}{2}\int \sqrt{\cos^2\theta}\ cos\theta\ d\theta$$
$$\frac{1}{2}\int cos^2\theta\ d\theta$$
Add an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Solution of $T(n)=2T(n/2) + n\log(\log n)$ I am struggling to solve this equation:
$$T(n)=2T(n/2) + n\log(\log n).$$ I concluded that the Master Theorem does not apply in this situation so I tried to successively substitute the terms in order to solve this equation but cannot proceed. Could somebody tell me what would ... | We are given that $$T(n) = 2 T(n/2) + n \log_2(\log_2(n))$$ (Note that if we take $\log$ to the base $2$ or to the base $e$, the only difference is in the coefficient of the leading order term).
Let $n = 2^k$. Call $T(n) = g(k)$. Then we have
\begin{align}
g(k) & = 2g(k-1) + 2^k \log_2(k)\\
& = 2 (2g(k-2) + 2^{k-1} \lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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What is wrong with my solution? $\int \cos^2 x \tan^3x dx$ I am trying to do this problem completely on my own but I can not get a proper answer for some reason
$$\begin{align}
\int \cos^2 x \tan^3x dx
&=\int \cos^2 x \frac{ \sin^3 x}{ \cos^3 x}dx\\
&=\int \frac{ \cos^2 x\sin^3 x}{ \cos^3 x}dx\\
&=\int \frac{ \sin^3 ... | Notice that:
$$
\frac{1}{2} \cos^2 x = \frac{1}{2} \left(\frac{1}{2} + \frac{1}{2} \cos 2x\right) = \frac{1}{4} + \frac{1}{4} \cos 2x
$$
And:
$$
-\ln|\cos x| = \ln|(\cos x)^{-1}| = \ln|\sec x|
$$
So your answer is correct. It just differs by a constant from the answer you expect.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/155829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 4
} |
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
| If $(1+x)(1+y)=2$, then
$$\begin{align}
x&=\frac{1-y}{1+y}\\
1+x^2&=2\frac{1+y^2}{(1+y)^2}\\
\frac{1+x^2}{1+x}&=\frac{1+y^2}{1+y}
\end{align}\tag{1}
$$
and since $(1+y)\,\mathrm{d}x+(1+x)\,\mathrm{d}y=0$ we get
$$
\frac{\mathrm{d}x}{1+x^2}=-\frac{\mathrm{d}y}{1+y^2}\tag{2}
$$
Therefore,
$$
\begin{align}
\int_0^1\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
"answer_count": 8,
"answer_id": 2
} |
Centroid of a region $$y = x^3, x + y = 2, y = 0$$
I am suppose to find the centroid bounded by those curves. I have no idea how to do this, it isn't really explained well in my book and the places I have looked online do not help either.
| The region you are interested is the blue shaded region shown in the figure below.
The coordinates of the centroid denoted as $(x_c,y_c)$ is given as $$x_c = \dfrac{\displaystyle \int_R x dy dx}{\displaystyle \int_R dy dx}$$ $$y_c = \dfrac{\displaystyle \int_R y dy dx}{\displaystyle \int_R dy dx}$$
where $R$ is the bl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$ I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
| Let $$I=\int\frac{dx}{x^4+1}$$
Enforce the substitution $x:=\frac{1}{y}\implies dx=-\frac{dy}{y^2}$ so that $$I=-\int\frac{dy}{y^2\left(\frac{1}{y^4}+1\right)}=-\int\frac{dy}{y^2+\frac{1}{y^2}}=-\frac{1}{2}\int\frac{1-\frac{1}{y^2}+1+\frac{1}{y^2}}{y^2+\frac{1}{y^2}}dy\tag1$$
Then observe that $$y^2+\frac{1}{y^2}=\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 20,
"answer_id": 19
} |
Discriminant for $x^n+bx+c$ The ratio of the unsigned coefficients for the discriminants of $x^n+bx+c$ for $n=2$ to $5$ follow a simple pattern:
$$\left (\frac{2^2}{1^1},\frac{3^3}{2^2},\frac{4^4}{3^3},\frac{5^5}{4^4} \right )=\left ( \frac{4}{1},\frac{27}{4},\frac{256}{27},\frac{3125}{256} \right )$$
corresponding to ... | Use the relation between the disciminant of $f$ and the resultant of $f$ and $f'$. The resultant is easy to calculate since $f'$ is so simple.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/161530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Prove that $K$ is a field Let be $K$ the set of real numbers that can be written as $a+b\sqrt2$, with $a$ and $b$ rational numbers. Prove that $K$ is a field.
I have already proved that $0$ and $1$ $\in K$, and that sum and product of two elements $\in K$. I have also already proved that the opposite $\in K$. I don't k... | Let $a+b\sqrt{2}\in K$ and $a+b\sqrt{2}\neq 0$. If $b=0$, then $a+b\sqrt{2}=a\neq 0$ and $1/a$ is its multiplicative inverse. Therefore, we assume $b\neq 0$, which implies that
$a^2-2b^2\neq 0$. Otherwise, if $a^2-2b^2=0$, then $\sqrt{2}=\frac{a}{b}$ which is rational since $a, b$ are rational, which contradicts to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Simplest method to find $5^{20}$ modulo $61$ What is the simplest method to go about finding the remainder of $5^{20}$ divided by $61$?
| $5^3=125=2\cdot61+3$, so $5^3\equiv 3\pmod{61}$. $3^5=243=4\cdot61-1$, so $5^{15}\equiv 3^5\equiv-1\pmod{61}$. Finally, $5^{20}=5^{15}\cdot5^3\cdot5^2\equiv-1\cdot3\cdot25\equiv-75\equiv-14\equiv47\pmod{61}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/163186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 4
} |
Factoring $x^4z-2z^2-4x^6+x^2z$ We want to factor $8x^4y^4-2y^8-4x^6+x^2y^4 = -2y^8 + (8x^4+x^2)y^4 -4x^6$. We substitute $x^4$ with $z$:
Now we want to compute this $8x^4z-2z^2-4x^6+x^2z = -(x^2-2z)(4x^4-z)$ by hand.
Therefore we transform it into $-(2z^2-(8x^4+x^4)z+4x^6z^0)$ and use the quadratic formula on the (inn... | $$8x^4y^4-2y^8-4x^6+x^2y^4 =(8x^4y^4-2y^8)-(4x^6-x^2y^4)=2y^4(4x^4-y^4)-x^2(4x^4-y^4)=$$
$$=(2y^4-x^2)(4x^4-y^4)=(2y^4-x^2)((2x^2)^2-(y^2)^2)=(2y^4-x^2)(2x^2-y^2)(2x^2+y^2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/163539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find $\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$ $$\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$$
I have tried multiplying by $\frac{1}{\sqrt{x^2+4}}$ and it's reciprocal, but I cannot seem to find the solution. L'Hospital's doesn't seem to work either, as I keep getting rational square roots.
| Hint $$\mathop {\lim }\limits_{x \to \infty } \dfrac{{\sqrt {{x^2} + 4} }}{{x + 4}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\dfrac{{\sqrt {{x^2} + 4} }}{x}}}{{\dfrac{{x + 4}}{x}}} = \mathop {\lim }\limits_{x \to \infty } \dfrac{{\sqrt {\dfrac{{{x^2} + 4}}{{{x^2}}}} }}{{1 + \dfrac{4}{x}}} = \mathop {\lim }\limit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Two proofs I'm having difficulty with I've been given an assignment. Almost done except the last two are tripping me up. They are as follows:
1) if $2x^2-x=2y^2-y$ then $x=y$
2) if $x^3+x=y^3+y$ then $x=y$
I imagine they use a similar tactic as they both involve powers, but I've tried factoring,completing the square, d... | For 1) Observe that (I am assuming $x,y$ are real numbers):
\begin{align}
2x^2-x=2y^2-y\\
&\implies (2x^2-x)-2y^2+x=(2y^2-y)-2y^2+x\\
&\implies 2(x^2-y^2)=(x-y)\\
&\implies 2(x-y)(x+y)=(x-y)\\
&\implies 2(x-y)(x+y)-(x-y)=(x-y)-(x-y)=0\\
&\implies 2(x-y)(x+y)-(x-y)=0\\
&\implies (x-y)(2(x+y)-1)=0\\
&\implies (x-y)=0 \te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Prove that there exists a natural number n for which $11\mid (2^{n} - 1)$ I'm thinking putting it into modulo form: there exists a natural number $n$ for which
$$2^{n}\equiv 1 \pmod {11}$$
but I don't know what to do next and I'm still confused how to figure out remainders when doing modulos, like $2^n\equiv \;?? \pmod... | A natural number $n$ will have the property that $11\mid 2^n-1$ precisely when $n$ is a multiple of $10$.
$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
n & \!\!\!\!\!& 1 & 2& 3& 4& 5& 6& 7&8&9&\mathbf{\Large 10}&11&12&13&14\\\hline\\
2^n\bmod 11 & \!\!\!\!\!& 2 & 4& 8 &5 &10 &9 & 7&3 &6&\mathbf{\Large 1 }&2&4&8&5
\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers I have tried the arithmetic-geometric inequality on $(-a+b+c)(a-b+c)(a+b-c)$ which gives
$$(-a+b+c)(a-b+c)(a+b-c) \leq \left(\frac{a+b+c}{3}\right)^3$$
and on $abc$ which gives
$$abc \leq \left(\frac{a+b+c}{3}\right)^3.$$
Since ... | A proof using the order inequality:
As @robjon points out, no two of $-a+b+c, a-b+c, a+b-c$ can be negative.
If exactly one of them is negative, the inequality is trivial.
Hence, assume that $-a+b+c, a-b+c, a+b-c > 0$.
Now, it's not difficult to see that $(b+c, a+c, a+b)$ and $(-a, -b, -c)$ are equally ordered.
And a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 7,
"answer_id": 6
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Complete the square and write in standard form for $3x^2+3x+2y=0$
Standard forms: $y-b=A(x-a)^2$ or $x-a=A(y-b)^2$
$3x^2+3x+2y=0$
I honestly do not know how to start this problem. I have tried a lot of things and obviously not the right one. Can someone explain to me the first step and nothing more and I will ed... | Since you have a factor with $x^2$, this anticipates the form you will be aiming for is
$$y-b=A(x-a)^2$$
So let's look at your eqn.:
$$3x^2+3x+2y=0$$
We need to produce a perfect square with $3x^2+3x$. So, we can do the old completing the square trick:
$$3x^2+3x=3(x^2+x)$$
$$3x^2+3x=3(x^2+2\frac 1 2 x)$$
$$3x^2+3x=3\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove $\frac{\sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{\tan A}{1-\tan^2 A}$ How would I simplify this difficult trigonometric identity:
$$\frac{\sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{\tan A}{1-\tan^2 A}.$$
I am not exactly sure what to do.
I simplified the right side to
$$\frac{\frac{\sin A}{\cos A}}{1-\frac{\c... | Use $\sin 2A= 2\sin A \cos A$ and
$ \cos 2A= \cos^2 A- \sin^2 A$ to get
$$
\frac{\sin A \cos A}{\cos^2 A- \sin^2 A}=\frac{\sin 2A}{2\cos 2A}=\frac12\tan{2A},
$$
which is equivalent to
$$
\frac{ \tan A} {1 - \tan^2 A} .
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/171692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Sum of the series : $1 + 2+ 4 + 7 + 11 +\cdots$ I got a question which says
$$ 1 + \frac {2}{7} + \frac{4}{7^2} + \frac{7}{7^3} + \frac{11}{7^4} + \cdots$$
I got the solution by dividing by $7$ and subtracting it from original sum. Repeated for two times.(Suggest me if any other better way of doing this).
However now ... | (Edit: Upps, I see now this is essentially solution (2) of Peter Tamaroff's answer, but because it's much shorter I just leave it here)
Your sequence can be separated into 2 sequences, where we add each pair:
$\begin{eqnarray}
&1&2&4&7&11&16&\cdots & = a_k\\
\hline
=&1&1&1&1&1&1&\cdots \\
+&0&1&3&6&10&15&\cdots \\
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 5
} |
multiple choice matrices problem If $M$ is a $3 \times 3$ matrix such that
$$
\begin{aligned}
\begin{pmatrix} 0 & 1 &2\end{pmatrix}M &= \begin{pmatrix} 1 & 0 &0\end{pmatrix} \text{ and}\\
\begin{pmatrix} 3 & 4 &5\end{pmatrix}M &= \begin{pmatrix} 0 & 1 &0\end{pmatrix} \text{ ,}
\end{aligned}
$$
then $\begin{pmatrix} 6 &... | Considering the Options,we have $\begin{pmatrix} 6 &7 &8\end{pmatrix}M=\begin{pmatrix} -1 &2 &0\end{pmatrix}$. Since If $A$ be an $m\times n$ matrix and we partition $A$ into rows then $$A = \begin{bmatrix}
a(1,:) \\ a(2,:) \\ a(3,:)
\end{bmatrix}$$ If $B$ an $n\times r$ matrix, then the $i$th row of product $AB$ is d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral Here is a fun integral I am trying to evaluate:
$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$
I thought about integrating by parts $2n$ times and then using the binomial theorem for $\sin(x)$, that is, us... | Using
$$
\sin^{2n+1}(x) = \sum_{k=0}^n \frac{(-1)^k }{4^n} \binom{2n+1}{n+k+1} \sin\left((2k+1)x\right)
$$
We get
$$ \begin{eqnarray}
\int_0^\infty \frac{\sin^{2n+1}(x)}{x}\mathrm{d} x &=& \sum_{k=0}^n \frac{(-1)^k }{4^n} \binom{2n+1}{n+k+1}\int_0^\infty \frac{\sin\left((2k+1)x\right)}{x}\mathrm{d} x\\ &=& \sum_{k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 2
} |
Prove trigonometry identity for $\cos A+\cos B+\cos C$ I humbly ask for help in the following problem.
If
\begin{equation}
A+B+C=180
\end{equation}
Then prove
\begin{equation}
\cos A+\cos B+\cos C=1+4\sin(A/2)\sin(B/2)\sin(C/2)
\end{equation}
How would I begin the problem I mean I think $\cos C $ can be $\cos(180-A+B)... | What I might do is start with the right side. Since I don't remember half-angle formulas, let $a = A/2$, $b=B/2$. Note that (from the addition formulas for $\cos$) $\sin(x) \sin(y) = (\cos(x-y) - \cos(x+y))/2$, $\cos(x) \cos(y) = (\cos(x+y) + \cos(x-y))/2$, $\sin(90 - x) = \cos(x)$.
$$ \eqalign{1 &{}+ 4 \sin(a) \sin(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Gre Question Complex Number (plug and chug) This seems like it should be easy, but I can't seem to simplify it: If $z=e^{i\frac{2\pi}{5}}$, then what is $1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9$. The choices are $0, 4e^{i\frac{3\pi}{5}}, 5e^{i\frac{4\pi}{5}}, -4e^{i\frac{-2\pi}{5}}, -5e^{i\frac{3\pi}{5}},$ with the a... | Hint $\rm\ \ z\ne 1,\,\ z^5 = 1\:\Rightarrow\: (\color{#C00}{1\!+\!z\!+\!z^2\!+\!z^3\!+\!z^4})(1\!+\!4z^4)+5z^9 =\, \color{#C00}0+5z^9 =\, 5/z$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/179804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
what is the easiest way to represent $ \sqrt{1 + x} $ in series How to expand $ \sqrt{1 + x}$.
$$ \sum_{n = 0}^\infty {{\left ( 1 \over 2\right )!}x^n \over n! \left({1 \over 2 }- n\right )!} = 1 + \sum_{n = 1}^\infty {{\left ( 1 \over 2\right )!}x^n \over n! \left({1 \over 2 }- n\right )!}$$
How can I simplify $ \le... | Just to motivate a more elementary solution.
Let $$f(x)=\sqrt{x+1}=(x+1)^{1/2}$$
Now, we expand around $x=0$.
$$f'(x)=\frac 1 2 (x+1)^{1/2-1}$$
$$f''(x)=\frac 1 2 \left(\frac 1 2-1 \right)(x+1)^{1/2-2}$$
$$f'''(x)=\frac 1 2 \left(\frac 1 2-1 \right)\left(\frac 1 2-2 \right)(x+1)^{1/2-3}$$
By induction, we get
$$f^{(n)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Subsets and Splits
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.