Tagged Questions

1
vote
2answers
87 views

Can anyone help me understand this Strong Mathematical Induction proof?

I'm not sure if we're allowed to post pictures but I thought it would be easier to read and I didn't see anything in the rules about it. It's question 1. Section 5.4 This question: Here is the ...
1
vote
5answers
58 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
4answers
111 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
13
votes
4answers
574 views

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ [duplicate]

Yesterday, my uncle asked me this question: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$. How can we do this? Note that this is not a diophantine ...
1
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2answers
58 views

Polynomial Rewriting Proof

Note. Please provide only a hint along with some explanation, but not the answer. I want to struggle with this problem. This is not homework. Show that for any number $c$, a polynomial $ P(x) = ...
0
votes
6answers
129 views

Show that $ 2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4) = (a+b+c)(-a+b+c)(a-b+c)(a+b-c) $

I was trying to prove the Heron's Formula myself. I came to the expression $ 2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4) $ ). In the next step, I have to find $ (a+b+c)(-a+b+c)(a-b+c)(a+b-c) $ from it. But ...
2
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2answers
131 views

Simple proof; if $x$ is odd then $x^2 -1$ is divisible by 8

I have this. $x = 2k+1 $ $(2k+1)^2 -1 = 4k^2 +4k + 1 -1 $ $\frac{4k^2 +4k}{8} = \frac{k(k+1)}{2}$ At the end part I can see that for what $k$ is, the number on top is divisible by 2. I was expecting ...
0
votes
0answers
115 views

How to prove that $(1^3+2^3+\cdots+n^3)=(1+2+\cdots+n)^2$ [duplicate]

Possible Duplicate: Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ How can one prove that ...
5
votes
3answers
167 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: ...
1
vote
2answers
87 views

Theorem about two real numbers 2

My question is: Prove- If $a,b$ are two positive real numbers such that their sum is $a+b=k$. Then the product $ab$ is maximum if and only if $a=b=\displaystyle\frac{k}{2}$. I proved the ...
7
votes
4answers
182 views

Theorem about two real numbers

My question is: $a,b$ are two positive real numbers such that their product is constant,equal to $k$ say. Prove: the sum $a+b$ is minimum if and only if $a = b= \sqrt k$. Can this be solved using ...
1
vote
4answers
273 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 ...
4
votes
5answers
339 views

Showing $a^2 < b^2$, if $0 < a < b$

Lately, I've been stumbling with proofs of inequalities. For example: Given $0 < a < b$ Show $a^2 < b^2$ The only thing I've been able to come up with so far: $a^2 < b^2$ ...