2
votes
1answer
38 views

An inequality for a quotient of polynomials

I am trying to prove the following to be true for $n > 1$: $$\frac{n^4}{n^3 + 1} \le Cn$$ It seems like there is some basic rule where you multiply the 1 in the denominator by a value which makes ...
0
votes
1answer
66 views

Show without expanding that the two determinants are equal

$$ Let\ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
3
votes
0answers
60 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
0
votes
1answer
57 views

My proof is wrong, can anyone tell me why?

$$\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}, [x(x+1) = y(y+1)] \Leftrightarrow [x = y]$$ $$\forall x \in \mathbb{Z} , \forall y \in \mathbb{Z}, [x(x+1)=y(y+1)]\Leftrightarrow [x=y]$$ ...
5
votes
5answers
167 views

Show that $\frac{\sqrt{8-4\sqrt3}}{\sqrt[3]{12\sqrt3-20}} =2^\frac{1}{6}$

This was the result of evaluating an integral by two different methods. The RHS was obtained by making a substitution, the LHS was obtained using trigonometric identity's and partial fractions. Now I ...
2
votes
4answers
89 views

Easier Proof of $\sin{3\theta} + \sin\theta = 2\sin{2\theta}\cos\theta$

I am curious to see whether anybody can give me a proof that takes less steps. Here is how I did it: $$\sin{3\theta} + \sin\theta = 2\sin{2\theta}\cos\theta$$ LHS $$\eqalign{\sin(2\theta + \theta) ...
2
votes
3answers
138 views

Prove the distributive law $a(b+c)=ab+ac$ for real numbers?

I have always taken these kinds of things for granted. Well of course $a(b+c)=ab+ac$! But why? The thought randomly popped in my head, and I realized that I could not prove it. Perhaps we should take ...
0
votes
3answers
98 views

Absolute values don't work

I don't understand, how absolute valued could possibly be considered well defined. As shown here, $|a| = |-a| , ||a|| = |-|a||$ So lets take $a=-2, |a| = -2 = |-a|,$ but $|-a| = |2| = 2$ But it ...
5
votes
0answers
59 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
0
votes
1answer
31 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
0
votes
3answers
41 views

Let $x,y \in \mathbb{R}_{>0}$. If $x<y$ then $0<1/y<1/x$.

I came across a proof in my textbook and was wondering how to solve it: Let $x,y \in \mathbb{R}_{>0}$. If $x<y$ then $0<1/y<1/x$.
1
vote
3answers
105 views

Prove that $\sec^2{\theta}=(4xy)/(x+y)^2$ only when $x=y$

Show that the equation below is only possible when $x=y$ $$ \sec^2{\theta}=\frac{4xy}{(x+y)^2}$$ The only way I can think of doing this is by rewriting it as $$ ...
0
votes
1answer
25 views

Floor and Ceiling question

This was a homework question. I wasn't able to get far because I couldn't determine the properties of floor and ceiling functions. Any help would be awesome. $\def\lc{\left\lceil} ...
4
votes
3answers
137 views

Proof: $ \lfloor \sqrt{ \lfloor x\rfloor } \rfloor = \lfloor\sqrt{x}\rfloor $.

I need some help with the following proof: $ \lfloor \sqrt{ \lfloor x\rfloor } \rfloor = \lfloor\sqrt{x}\rfloor $. I got: (1) $[ \sqrt{x} ] \le \sqrt{x} < [\sqrt{x}] + 1 $ (by definition?). (2) ...
0
votes
4answers
55 views

if x^2 + 2x - 3 >= 0 then (x <= -3) V (x >= 1)

I know why this is true but putting it in symbolic notation has me stumped. so basically i have that: ...
0
votes
1answer
64 views

Proving $\left\lfloor n\frac{\log (b)}{\log (a)}\right\rfloor =\left\lfloor \frac{\log \left(b^n+1\right)}{\log (a)}\right\rfloor$

Inspired by this question, I'd like to know how one would go about proving the below more general equation? $$n \in \mathbb{N},\;a \in \mathbb{N},\;b \in \mathbb{N}$$ $$b^n+1 \notin ...
0
votes
1answer
74 views

Easy Proofs with Functions and Big-O

I have these two questions. I tried answering them, but got them wrong and I don't know how to answer them correctly. This is not homework --- I'd appreciate a solution (at least to one), and an ...
2
votes
4answers
141 views

prove by induction: $3 + 5 + 7 + … + (2n+1) = n(n+2)$

Use the principle of mathematical induction to prove that $$3 + 5 + 7 + ... + (2n+1) = n(n+2)$$ for all n in $\mathbb N$. I have a problem with induction. If anyone can give me a little insight ...
0
votes
3answers
58 views

For $x+y+z=0$, if $x$ and $y$ are divisible by some integer $k$, then so is $z$.

If k|x and k|y and x+y+z = 0, then k|z. Here, "k|x" means that $k$ is a divisor of $x$ and $x,y,z,k \in \mathbb{Z}$ What strategy would you employ to prove this?
0
votes
2answers
43 views

How to prove this statement and its negation?

Assuming that you're dealing with real numbers, d ^ 2 = e ^ 2, then d = e Why would it be true? << corrected, it is not true! thanks to posters What is the negation and is it true?
4
votes
1answer
255 views

Problems with fake proofs of limit of sequences

I can hardly imagine an easier example of the fact that my understanding of the topic is more than rusty. I will divide the question in two parts to make the reading easier: 1) Background; 2) ...
1
vote
3answers
113 views

Arithmetic and geometric mean

I need to prove that for $a=\frac{x+y}{2}$ and $g=\sqrt{xy}$, following statments are true or false: For $x\not =y,a>g$ and $x=y, a=g$. I have no idea how to do this, so any help is welcomed. ...
0
votes
2answers
166 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
0
votes
3answers
114 views

Prove that a circle has an infinite number of tangents

It seems obvious that a circle is comprised of the set of all points that are equidistant from one point, and that each point on the circumference of the circle represents a tangent. This seems to ...
7
votes
3answers
171 views

If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:

If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then: $$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) ...
1
vote
2answers
137 views

Show that $\frac{a+b}{2} \ge \sqrt{ab}$ for $0 \lt a \le b$

I have to prove that $$\frac{a+b}{2} \ge \sqrt{ab} \quad \text{for} \quad 0 \lt a \le b$$ The main issue I am having is determining when the proof is complete (mind you, this is my first time). So I ...
2
votes
1answer
59 views

Please help me to prove this statement

Prove that $$(3x^2 −7x−2012)(3x^2 −7x−2011)(3x^2 −7x−2010)(3x^2 −7x−2009)+1$$ is equal to a number squared. I first thought to multiply all, but I was stucked with big numbers, so I quit. Thx for ...
1
vote
2answers
684 views

Sum of absolute values and the absolute value of the sum of these values?

I'm working on a proof and I need some help with this: I determined that for some situations ($x$ or $y$ are negative but not both): $|x| + |y| > x + y$ How can I conclude using that statement ...
3
votes
2answers
67 views

For what $(m, n)$, does $1+x+x^2 +\dots+x^m | 1 + x^n + x^{2n}+\dots+x^{mn}$?

For what $(m, n)$, does $1+x+x^2 +\dots+x^m | 1 + x^n + x^{2n}+\dots+x^{mn}$? Well, $$\sum_{i = 0}^{m} x^i = \frac{x^{m+1} - 1}{x - 1}$$ and, $$\sum_{i = 0}^m x^{in} = \frac{x^{n(m+1)} - ...
0
votes
3answers
109 views

Help to understand and complete a proof by induction, $a^n < b^n$

I want to check if I understand proof by induction, so I want to proof the following: $a^n<b^n$ for $a,b \in \mathbb{R}$, $0<a<b$, $n \in \mathbb{N}$ and $n>0$ Here's my attempt: ...
2
votes
3answers
189 views

Algebra Textbook

Perhaps this questions was asked already, but I browsed through other threads and couldn't find exactly what I am looking for. I am looking for an Algebra Textbook (high-school/undergrad level) that ...
3
votes
1answer
301 views

Why n! equals sum of some expression?

Why n! equals sum of some expression? Especially I need to know why this expression is true? $$ n!= \left(\frac{n+1}{2}\right)^{p(n)} \; \prod_{j=0}^{q(n)}\sum_{i=0}^j(n-2i), $$ Where \begin{gather*} ...
16
votes
5answers
407 views

Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$

How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$ Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't ...
2
votes
2answers
104 views

Whats wrong with this proof?

Theorem: $x$ is a real number with $x \neq 1.$ If $\frac {x^2+1}{x-1} =x$, then $x=-1$. If we suppose that $x=-1$. Then $\frac {x^2+1}{x-1} = \frac {(-1)^2+1}{-1-1} = \frac {2}{-2} = -1 = x$ I would ...
7
votes
3answers
221 views

How to understand proof of a limit of a function?

Given the following function: $$ f(x)=\left\{ \begin{array} {cc} 0, & x \text{ irrational, } 0<x<1 \\ \frac{1}{q}, & x=\frac{p}{q} \text{ in lowest terms, } 0<x<1 \end{array} ...
1
vote
2answers
168 views

I don't understand this proof of the AM-GM inequality?

The proof uses this lemma which I understand: $\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ ...
1
vote
1answer
971 views

Suppose a,b are real numbers, if a is rational and ab is irrational, then b is irrational (Is my solution correct?)

Suppose $a,b$ are real numbers, if $a$ is rational and $ab$ is irrational, then $b$ is irrational. Solution: Proof by contraposition $$b = \frac{p}{q}$$ $$ a = \frac{j}{k}$$ where $p,q,j,k$ are ...
6
votes
4answers
162 views

Is there a simpler approach to these system of equations?

I recently came across the following system of equations: $$x + y + z = 1 \\ x^2 + y^2 + z^2 = 2 \\ x^3 + y ^3 + z^3 = 3$$ And I have two questions: One, is there a way to prove or disprove ...
7
votes
2answers
386 views

How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
1
vote
2answers
187 views

Prove the following ceiling and floor identities?

Could someone help me prove these identities? I'm completely lost: $$\begin{align*} &(1)\quad \left\lceil \frac{\left\lceil \frac{x}{a} \right\rceil} {b}\right\rceil = \left\lceil ...
2
votes
6answers
156 views

Prove that $\sum_{j = 0}^{n} (-\frac{1}{2})^j = \frac{2^{n+1} + (-1)^n}{3 \times 2^n}$ whenever $n$ is a nonnegative integer.

I'm having a really hard time with the algebra in this proof. I'm supposed to use mathematical induction (which is simple enough), but I just don't see how to make the algebra work. $\sum_{j = 0}^{k} ...
4
votes
1answer
202 views

Basic Proof Question

I'm working on a basic proof for my intro proof course. The text is Analysis with an Introduction to Proof by Lay. This question comes from section 2. I am asked to proof or disprove the following: ...
1
vote
3answers
115 views

How to prove $1 + a + \cdots + a^n + \cdots = \frac{1}{1-a}$? [duplicate]

Possible Duplicate: Value of $\sum\limits_n x^n$ Let $\lvert a \rvert < 1$. How would I show that the infinite sum $$1 + a + \cdots + a^n + \cdots = \frac{1}{1-a}$$ Update: Thanks ...
0
votes
1answer
104 views

Not following algebra in a proof, can anyone please explain it?

So I'm learning mathematical induction as a proof technique (teaching myself discrete math as a foundation for a comp sci class I'm going to be taking). My algebra is a little rusty, and I cannot ...
4
votes
1answer
256 views

Invariants in a second order equation

For $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$, why are $\begin{vmatrix} A &B \\ B &C \end{vmatrix}$ and $\begin{vmatrix} A & B & D\\ B & C & E\\ D & E & F \end{vmatrix}$ ...
1
vote
2answers
92 views

Show that if $\lfloor x+a \rfloor$ = $\lfloor x+b \rfloor, \forall x \in \Bbb R$ then $a=b$; is showing that $x+a=x+b$ enough?

If i show that $x+a=x+b$ only if $a=b$, does that prove that the above is also true? $ x+a=x+b \iff x+a-x-b=0 \iff a-b=0 \implies b=a$ also is this any good?
5
votes
3answers
207 views

How to show x and y are equal?

I'm working on a proof to show that f: $\mathbb{R} \to \mathbb{R}$ for an $f$ defined as $f(x) = x^3 - 6x^2 + 12x - 7$ is injective. Here is the general outline of the proof as I have it right now: ...
5
votes
5answers
472 views

How to disprove there exists a real number $x$ with $x^2 < x < x^3$

I realize that the only method is to show various cases: I must test for $x > 1$, $x < -1$, $0 \leq x \leq 1$, and $-1\leq x \leq0$. But even with this, I don't understand how to inject the ...
4
votes
3answers
329 views

How to prove floor identities?

I'm trying to prove rigorously the following: $\lfloor x/a/b \rfloor$ = $\lfloor \lfloor x/a \rfloor /b \rfloor$ for $a,b>1$ So far I haven't gotten far. It's enough to prove this instead ...
1
vote
4answers
364 views

Proof by Contradiction Problem Where do i start

Prove the following: There are no rational number solutions to the equation $x^3 +x+ 1$ = 0, i.e. no solution can be written as a ratio a/b where a, b ∈ N (you can always consider a/b to be reduced to ...