# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $0<1/y<1/x$.
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### 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 ...
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### 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 ...
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### 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?
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### 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?
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### 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) ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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*} ...
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### 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 ...
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### 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 ...