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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0
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1answer
15 views

Algebraic manipulation and logarithms

How can i manipulate 3*(n/2)^(log base 2 of 3) to equal n^((log(3))/(log(2)))? I understand that log base b of a = log base 2 of(a)/log base 2 of(b) but im not sure how the 3/2 went away.
-5
votes
1answer
13 views

is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$ and vice versa? [on hold]

Is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$? Also is there constant $k$ that $k2^n>F_n$?
2
votes
0answers
20 views

Extremal points relative to origin for an ellipsoid

Suppose I have an ellipsoid of the form $ax^2 + by^2 + az^2 - cxy -cyz = d$ How would I find the points nearest to, and furthest from, the origin?
0
votes
2answers
44 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
1
vote
2answers
55 views

Sum of $1/n+1/(n-2) + 1/(n-4) + \cdots $

How does one calculate $$\frac{1}{n} + \frac{1}{n-2} + \frac{1}{n-4} \cdots $$ where this series continues until denominator is no longer positive? $n$ is some fixed constant positive integer.
1
vote
2answers
31 views

polynomial of $4^\text{th}$ degree, prove

There is a polynomial $f$ of integer coefficients such that $\text{deg(f)} \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no ...
1
vote
2answers
90 views

Proving a function is onto?

Let $f: \mathbb{R}\setminus \{3\} \to \mathbb{R}\setminus \{1\}$ be defined by $f(x)=\dfrac{x+3}{x-3}$ Prove that $f$ is onto: Okay, here is the deal. I just started my first abstract algebra ...
0
votes
2answers
22 views

Manipulating an expression into another equivalent form

I have an expression (shown below) and I want to show that $$(n+1)(n)(3n^2+11n+10) = (n)(n+1)(n-1)(3n+2) + \text{some other stuff}$$ How can I do this?
-3
votes
1answer
21 views

…is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences. [on hold]

Let $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that $c\phi^n + ...
0
votes
3answers
37 views

Why $|x-y|<1\implies|y|\leq |x|+1$?

I have the following passage in one of the proofs in my workbook: $$|x-y|<1\implies|y|\leq |x|+1$$ Why is this valid?
0
votes
1answer
17 views

Multiplying brackets in $n(n+1)/2+n+1$

Why does: $$n(n+1)/2+n+1 = (n^2+3n+2)/2 $$ and not $$ (n^2+2n+1)/2 $$ ? Additionally, why is: $$(n^2+3n+2)/2 = ((n+1)(n+1)+1)/2$$ rather than: $$((n+1)(n+1)+1n)/2$$
0
votes
0answers
8 views

How to calculate recurrence $F(n) = F(n/u) + \Theta(n^k)$ where $u,k \in \mathbb{N}$

$\Theta$ is used as in Bachmann-Landau notation (often called as Big-O notation convention). How does one in general the recurrence relation of the following from: $$F(n) = F(n/u) + \Theta(n^k) ...
1
vote
1answer
10 views

How to calculate direct proportionality with logarithms and constant terms added

For the equation: $$y=a-b-c\log(x)$$ How do I calculate how $y$ scales with $x$? This is simple without the logarithms. For example: $$y=a+bx$$ $$y=b(\frac{a}{b}+x)$$ $$y\propto(\frac{a}{b}+x)$$ ...
0
votes
0answers
29 views

Cubic curve with a point of inflection

Not quite what I wanted to ask. What I really wanted to know is why you can't have a cubic curve that starts from top left and ends top right.
5
votes
0answers
39 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
-4
votes
1answer
17 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? [on hold]

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.)
0
votes
1answer
67 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
vote
1answer
34 views

How can I solve a system of two equations, like $A + B = 13$ and $2D + B = 13$?

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
vote
2answers
30 views

Factorial formula problem [duplicate]

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
0
votes
1answer
13 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
votes
2answers
23 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
votes
2answers
28 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 ...
3
votes
1answer
51 views

Relationship between increasing integer sequences

Suppose that $\mathcal X\cap \mathcal Y=\emptyset$, that $\mathcal X\cup \mathcal Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to \mathcal X$ respectively $\Bbb N\to \mathcal ...
1
vote
1answer
21 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
votes
0answers
39 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 ...
10
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0answers
66 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
votes
0answers
29 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
votes
2answers
21 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
votes
2answers
57 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
votes
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 $an + b$.
-6
votes
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?
1
vote
1answer
37 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 $n$.
0
votes
1answer
30 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
vote
2answers
75 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
vote
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
45 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
19 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
vote
0answers
40 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 ...
18
votes
7answers
2k 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
vote
1answer
17 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
votes
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 ...
-2
votes
1answer
21 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
30 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
votes
1answer
26 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
votes
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
vote
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 ...
-1
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