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
votes
4answers
25 views

Finding maximum of a function with unknown constants

I have a function in the form: $$y = \frac{ax}{b + \frac{x^2}{c} + x}$$ Supposedly, the maximum of this function is equal to $\sqrt{bc}$. I've tried substituting in $\sqrt{bc}$ for $x$, but I don't ...
0
votes
4answers
50 views

How many solutions has this third degree equation?

how many solutions has this equation: $$ {x}^{3}+4\,{x}^{2}-1=0 $$ i tried ruffini so far and it is not working, now i'm stuck and no idea of how to aproach this.
0
votes
0answers
18 views

Integer solutions to an equation

Let $x,y,z$ be positive integers and $S$ be the set of all the solutions to the equation $x^y+y^z=z^x$. Is $S$ finite or infinite? Lots of thanks for any help in advance.
0
votes
0answers
8 views

Find a cyclic rational function such that…

I'm looking for a function of the form $\frac{f(a,b,c)}{f(b,c,a)}$ (or close to this form, e.g. $\frac{(a+b)^2}{b^2+bc+c^2}$) which is roughly equal to $\frac{b^3-a^2-b^2-a^3-ab^2}{b^2c+a^2b+b^3}$ (I ...
2
votes
2answers
56 views

Prove, inequality ,positive numbers

$$\frac{a}{e+a+b}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{c+d+e}+\frac{e}{d+e+a}<2$$ Prove that for positive numbers $a,b,c,d,e$ there is such inequality
-1
votes
3answers
22 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...
2
votes
2answers
42 views

divide 6 people in group of 2 in same size

Exercise: divide 6 people in group of 2 in same size. My solution: The exercise tells us to calculate the combination without repetition. If I start by calculating the number of ways to select how ...
0
votes
1answer
25 views

Algebraic manipulation and logarithms

How can i manipulate $3\left(\dfrac{n}{2}\right)^{\log_2 3}$ to equal $n^{\frac{\log 3}{\log 2}}$? I understand that $$\log_b a = \dfrac{\log_2 a}{log_2 b}$$ but i'm not sure how the $3/2$ went away.
-5
votes
1answer
15 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
24 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?
-1
votes
2answers
51 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
61 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
37 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
97 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
22 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
38 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
18 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
9 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: $$y=a+bx$$ $$y=b(\frac{a}{b}+x)$$ $$y\propto(\frac{a}{b}+x)$$ ...
1
vote
1answer
37 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
42 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
2answers
76 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
31 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
14 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
26 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
53 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
42 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 ...
11
votes
1answer
82 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
31 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
34 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
65 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
31 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
61 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 ...
2
votes
3answers
53 views

Why does $\sqrt{x} / y =\sqrt{x/y/y}$?

Sorry for the awkward title, hard to to sum a mathematical problem with words alone. Having said that, I recently learned that the root of any value, $x$, and then that over value $y$, is identical ...
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 ...
29
votes
8answers
4k 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 ...