Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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0answers
12 views

How to find the sum of $\displaystyle S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ Hi! For this question, I did the following, Let $$ S=\sum_{n=1}^{\infty} \dfrac{\sin ...
-3
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0answers
10 views

If two people temporarily covered the cost of \$20 for the 3rd person by paying \$10 each, how much would the 3rd person owe person 1 and 2?

If two people temporarily covered the cost of 20 for the 3rd person by paying 10 each, how much would the 3rd person owe person 1 and 2? ( so that everyone is paying the same amount in the end.)
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1answer
20 views

A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is …

A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is $1 - \frac{a^{n+b}}{3^{2^{n+c}}-1}$, where $a$, $b$, and $c$ are integers. Find $a+b+c$.
1
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1answer
24 views

How can I solve this algebraic expression?

I am currently studying for my SSAT and this question appeared in my practice book: When $A + B = 13$ and $2D + B = 13$, what is the value of $D$? (A) 13 (B) 5 (C) -5 (D) -7 ...
1
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2answers
27 views

Factorial formula problem

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
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1answer
12 views

Values of $w$ while $y$ changes

I know this is very simple, but I just can't manage to find it. I have $w, y \in \mathbb{N}^*$. Assume that $0 < y < 255$ and $500 \ge w \ge 138$. This is for an animation controlled by the ...
0
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2answers
18 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
0
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2answers
17 views

Show using inequality of means that $a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$

Show using inequality of means that for $a>0$ and $n\in\mathbb{N}$: $$a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$$ I'm sure it's not that complicated, but I'm probably missing ...
2
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0answers
28 views

Relationship between increasing integer sequences

Suppose that $X\cap Y=\emptyset$, that $X\cup Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to X$ respectively $\Bbb N\to Y$. Further, suppose that there are straight line ...
1
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1answer
17 views

Transformation of an equation

How do you get from the left side to the right side in this equation? $$\frac{1+\sqrt{5}}{2} + 1 =\left(\frac{1+\sqrt{5}}{2}\right)^2$$
3
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0answers
35 views

How does one solve $y^y-x^x=x$ for $x$ as a function of $y$?

In order to find the answer to this question I started thinking that as a first step to obtain the first and second column, one would have to solve the equation: $$y^y-x^x=x$$ for $x$ as a function ...
8
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0answers
52 views

$P(z)=0$ iff $Q(z)=0$, $P(z)=1$ iff $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x$

Assume $P(x)$ and $Q(x)$ are polynomials with complex coefficients with degree greater than or equal to $1$ such that $P(z)=0$ if and only if $Q(z)=0$, $P(z)=1$ if and only if $Q(z)=1$. Prove that ...
0
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0answers
27 views

quadratic formula for polynomials with variable coefficients

I have trouble calculating equations like the one in last comment in the first answer; Solve system of 3 equations there are variable coefficients which I can calculate using quadratic formula - if ...
0
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2answers
19 views

Computing an academic grade when relative weights are changed

My grade is 88.6% (High B) and we get 80%(Assessment Grade) and 10%(Homework). My teacher is now making this 70%(Assessment Grade) and 30%(Homework). I have done all my homework 100% and I've been ...
2
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2answers
48 views

If 2 people pay 10 each, how much would a 3rd person have to pay to have an equal share?

If person 1 and 2 pay $\$10$ to equal $\$20$, how much would person 3 have to pay person 1 and 2 to become even? My solution: 20 divided by 3 is 6.66 so wouldn't the 3rd person just have to pay ...
2
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2answers
36 views

Find the number of children, given that the estate was divided evenly between them [on hold]

Problem of the Week at University of Waterloo: A man died leaving some money in his estate. All of this money was to be divided among his children in the following manner: $x$ to the first ...
-3
votes
3answers
33 views

The closed form sum of $12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]$… [on hold]

The closed form sum of $12 \left(1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right),n \geq 1$ is $n(n+1)(n+2)(an+b)$. Find $an + b$.
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1answer
64 views

you know root square of -1, what is the larger of the square? [on hold]

there is a square ABDC, $BD = \sqrt{-1}$ what is the value of AB=BC=DC=AD?
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1answer
36 views

Find polynomial f(n) such that for all integers $n$ $\geq 1$, we have

Find polynomial f(n) such that for all integers $n \geq 1$, we have $3\left( 1\cdot2 + 2\cdot3 + \ldots + n(n+1) \right) = f(n)$. Write f(n) as a polynomial with terms in descending order of $n$.
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1answer
29 views

How to solve $D=\sqrt{X^2+MX^2}$ for $X$?

How I to solve $D=\sqrt{X^2+MX^2}$ for $X$? With my rudimentary experience, I find myself incapable. I apologize for asking a question after asking a similar one previously (several days ...
1
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2answers
71 views

Solve system of 3 equations

$x+y+z=0$ $x^2+y^2+z^2=6ab$ $x^3+y^3+z^3=3(a^3+b^3)$ this is what i reasoned out so far; $xyz=a^3+b^3$ $x^2+zx+z^2=3ab$ $y^2+zy+z^2=3ab$ $x^2+xy+y^2=3ab$ $y^2=3ab+zx$ $x^2=3ab+zy$ ...
1
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1answer
59 views

Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$

Four different positive integers $a, b, c$, and $d$ are such that $a^2 + b^2 = c^2 + d^2$ What is the smallest possible value of $abcd$? I just need a few hints, nothing else. How should I begin? ...
0
votes
3answers
44 views

The meaning of dot centered vertically, as in $3\cdot 5$

I just went to a site to practise some maths as I am studying some maths on my OU course in computing and IT and I ran into some symbols that I did not understand The questions were solve ...
0
votes
1answer
17 views

Rearranging $ca^{b-1}/d^2$

I'm am try to rearrange $\frac{ca^{b-1}}{d^2}$ to $\large{\frac{c}{d^2a^{b-1}}}$ but I am having difficulty. I have tried times both top and bottom with various expressions such as $a^{b-1}$ but with ...
1
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0answers
38 views

Closed form for the summation $\sum_{k=1}^n\dfrac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
16
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7answers
1k views

Is there something special about 2015?

Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ...
1
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1answer
16 views

Show that $dn^{\beta}/(\epsilon n^{1/2})^2$ can be written as $d /(\epsilon^2)(n^{(1-\beta)})$

Show that $dn^{\beta}/(\epsilon n^{1/2})^2$ can be written as $d /(\epsilon^2)(n^{(1-\beta)})$ I have tried $d n^{\beta}/(\epsilon^2) (n^{5/2})$ and then $dn^{(\beta-5/2)}/\epsilon^2$ But the 5/2 is ...
0
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2answers
27 views

The range of $\frac{2^x-1}{2^x+1}$

I am trying to find the range of the function $\frac{2^x-1}{2^x+1}$. If we draw using a graph plotter we can see that the range is $-2<y<2$. To find the upper bound, I tried ...
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votes
1answer
20 views

Can someone show me how this algebraic expression is worked out fully?

I'm not sure how they went from $\frac{k2i(1)2i(2)}{\frac{d}{8}}$ to 32F? I'm weak in algebra so if anyone has any reccomendations how I can improve in manipulating equations or websites and ...
0
votes
1answer
28 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
2
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1answer
24 views

Symmetric and homogeneous three variable inequality with radicals.

While trying to solve a problem, I got the following inequality which appears correct, but I cannot prove. For positive $x, y, z$, $$\sum_{cyc} \frac{x}{y^2+z^2} \ge \sum_{cyc} ...
-4
votes
1answer
37 views

How to solve: Differentiate d/dx 7^-1/2 [on hold]

All the question says Differentiate d/dx 7^-1/2
-4
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0answers
31 views

Trigonometric math problem [on hold]

A camera is mounted at a point 3000 ft from the base of a rocket launching pad. The Rocket rises vertically when launched, and the camera's elevation angle is continually adjusted to follow the bottom ...
1
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1answer
42 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
0
votes
3answers
27 views

Beginner exponent/simplification question

Hey there I am having some trouble remembering all the old exponent rules and such, for example, $$ \frac{1}{(6+7^n) ^3} $$ How can I simplify this? I know that (7^n)^3 is the same as (7^3n), but ...
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votes
0answers
34 views

How does $\frac{x-(x+h)}{kx(x+h)}=\frac{-1}{x(x+k)}$

I'm sorry, I know this is very basic. But I'm getting lost somewhere in the algebra. :( Thank you
-1
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2answers
125 views

Is $x/x$ continuous at $0$? [on hold]

Just wondering, while studying limit, if $x\over x$ is continuous at $0$. $f(0)={0 \over 0}$ ,, but $x/x=1$. In this case, is it continuous at $0$?
1
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2answers
43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
2
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1answer
41 views

Polynomial prove exercise

$P(x)=x^n + a_1x^{n-1} +\dots+a_{n-1}x + 1$ with non-negative coefficients has $n$ real roots. Prove that $P(2)\ge 3n$ I don't have an idea how to do that, I'm in 4th grade high school, you don't have ...
0
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2answers
32 views

Sum of the coefficients of the expansion

Find the sum of the coefficients of the expansion: $$\frac{(1+x)\cdot(2+x^2)\cdot(3+x^3)...(103 + x^{103})}{103!}$$ The answer says let $x=1$, is this the way to go? Why not let $x=0$ ??
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5answers
83 views

$x=yx$. Can this statement be true when we don't know that $y=1$?

I am dealing with an equation which is saying that $yx=x$. On the other hand it is telling us that $\frac{x}{x}=1$ which connotes that $x=x$. Is it not absurd to say that $x=x=yx$ if $x\neq{0}$ and ...
5
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0answers
55 views
+50

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
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3answers
32 views

Prove that $a^2+b^2+c^2+d^2+e^2 > a(b+c+d+e)$

Prove that $a^2+b^2+c^2+d^2+e^2 > a(b+c+d+e)$ Seems to be easy but, cannot see the method right now. Tried adding known things like $a^2+b^2>=2ab$ and so on with other letters.Maybe I didn't ...
1
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7answers
52 views

Logarithms with an answer that is a fraction

How does log base $16$ of $32$ equal $1.25$? If we divide $32/16=2$ but then if we divide $2/16$ it doesn't come out to a whole number unlike with log base $2$ of $4$ where $4/2=2$ and $2/2=1$ I am ...
1
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1answer
24 views

Application of Dimensional Analysis Problem

It is given that the radius $R$, in meters, of the expansion of a liquid in the soil is given by $t$ (time elapsed since the liquid was released), the mass $M$ of the liquid released and of the ...
-1
votes
3answers
36 views

simplify the equation [on hold]

I need help simplifying this equation. It is a fraction just in case the way I formatted it doesn't turn out right. $$ \frac{(4x + 3)^{1/2} − (x + 6)(4x + 3)^{−1/2}}{(4x+3)} $$
1
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1answer
34 views

Expected value of prime lottery ticket

Below is a problem I think that I have solved correctly, but cannot seem to get the correct answer. Any help would be greatly appreciated. You pay $\$13.00$ for a ticket. When you buy a ticket, ...
2
votes
2answers
31 views

How to go about solving this inequality question?

$\cos(3x-\pi/3) \leq (1/2).$ Here is what I have done so far... Let $3x-\pi/3 = X$. So I need to solve $\cos(X) \leq 1/2$. Which is all $X$ from $\pi/3$ to $5\pi/3$, so-- $\pi/3 \leq X \leq 5\pi/3 ...
5
votes
4answers
213 views

Why is $\frac{\sqrt{x+1}-1}{x}$ equal to $\frac{1}{\sqrt{x+1}+1}$?

I'm working with the expression $$\frac{\sqrt{x+1} - 1}{x}.$$ According to Wolfram Alpha "alternate form" section (http://www.wolframalpha.com/input/?i=%28%28x%2B1%29%5E1%2F2-1%29%2Fx) it is equal to ...
0
votes
2answers
35 views

To find inverse of function [on hold]

Given $ f(x) = \begin{cases} 2x, & \text{if $x\in[0,1]$} \\ 8 - 2x, & \text{if $x\in [2,3)$} \end{cases} $ Then how to find inverse of f ?