0
votes
0answers
20 views

Proof that the $sqrt[k]{z}\, z \in \mathbb N$ counts the amount of numbers less than or equal to z with a $k-$exact power

Empirically, it can shown that $$\mathrm{Floor}[\sqrt[k]{z} ] \,, z \wedge k\in \mathbb N $$ is equal to the amount of numbers which have a $k-$exact root. For example, $\sqrt 36 = 6$ means that there ...
0
votes
1answer
32 views

How to prove $|q|\ge 1 \Rightarrow |a|\ge |d|$?

Let $a,d,q \in \mathbb{Z}$ and $a=dq$ How do I show that $|q| \ge 1 \Rightarrow |a| \ge |d|$? I've tried: $|q|\ge 1 \Rightarrow (q>1 \text{, if } q>0) \text { or } (-q>1 \text{, if } ...
0
votes
4answers
69 views

Prove that (integer + non-integer) never equals an integer.

My question is how do you prove that given an integer $x$ and a number $y$, the only way for $x + y$ to be an integer is if $y$ is also an integer. I can see how to prove by induction that an integer ...
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 $0<1/y<1/x$.
2
votes
0answers
94 views

Bretschneider-Brahmagupta-Heron Proof

Derive Bretschneider's formula, Brahmagupta's formula and Heron's formula in one memorable elegant proof. I ask this question merely to see the creativity of the MSE community when it comes to ...
1
vote
0answers
48 views

Proof by induction for all positive integers

$\sum_{j=1}^n j^2 = \frac{n(n+1)(2n + 1)}{6}$ So I did the obvious and plugged one to show that $1^2 = \frac{1(1+1)(2(1) + 1)}{6}$ Now I am trying to show that $1^2 + 2^2 + ... + n^2 + (n + 1)^2 = ...
1
vote
1answer
86 views

7) Prove that $2n-3 \leq 2^{n-2}$ for all $n \geq 5$ by mathematical induction [duplicate]

Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction I have to prove by mathematical induction that: $2n-3\leq 2^{n-2}$ , for all $n \geq 5$
0
votes
2answers
53 views

Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction

Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction I have to prove by mathematical induction that: $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ Thank you for the Review.
1
vote
2answers
683 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 ...
1
vote
2answers
112 views

Using integrals to prove that the mean of the sampling distribution is the population mean

Let the random variables $X_1, X_2, \dots X_n$ denote a random sample from a population. The sample mean of these random variables is: $\overline{X}=\frac{1}{n}\sum\limits_{i=1}^{n}X_i$ I would like ...
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
2answers
171 views

How to prove that $\cos\theta$ is even without using unit circle?

The proofs I have come across on showing that $\cos \theta$ is even is something like this: In a unit circle, $\cos\theta$ gives you the $x$ coordinate after traveling $\theta$ radians ...
2
votes
3answers
114 views

if $-1<a<1$ and $-1<b<1$ how to prove that $(a+b)/(1+ab) $is also always between $-1$ and $1$?

I am trying to prove closure (among other things) a on algebraic structure $(A,*)$ where: $$A=(-1,1) ⊂ R,\ a*b=(a+b)/(1+ab)$$ (note that "*" here is NOT multiplication) So I have to prove that if: ...
1
vote
1answer
36 views

can we interpret a sentence involving more than one 'iff' in the following way?

If there are a sequence of 'iff' in a sentence, can we make a conclusion from the sentence by dropping all the 'iff' that lie after the first 'iff' and drop all the statements between the first ...
3
votes
2answers
142 views

Prove that $2^n>2n$ for all integral values of n greater than 2 [duplicate]

Prove $2^n >2n$ for all integral values of n greater than 2. Let $p_n$ be the statement: $$2^n>2n\ \forall\ n\gt2$$ If the inequality is valid for $n=k$ where $k>2$: $$p_k: 2^k>2k$$ ...
2
votes
1answer
95 views

Help with understanding a proof that $f$ is bounded on $[a,b]$ (Spivak)

I need help on the proof of Theorem 7-2 in Spivak: If $f$ is continous on $[a,b]$, then $f$ is bounded above on $[a,b]$. So, the proof starts with this: Let $$A= \{x:a\le x \le b \text{ ...
6
votes
4answers
321 views

How to show that a limit cannot be another number?

Let: $$ G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right. $$ I can understand that the function is continuous at $x=0$ because: For ...
5
votes
1answer
106 views

Help to understand the proof of $ \lim \limits_{x\to 0^+} f \left(\frac{1}{x}\right)=\lim \limits_{x\to \infty}f(x)$

The following is an answer to the proof of $$ \lim \limits_{x\to 0^+}f\left( \frac{1}{x} \right)=\lim \limits_{x\to \infty}f(x)$$ If $l=\lim \limits_{x\to \infty}f(x)$, then for every ...
1
vote
2answers
37 views

How $\delta_1$ and $\delta_2$ for two different limits at $a$ can be read as $\delta=\text{min}(\delta_1,\delta_2)$?

I am having trouble understanding a certain part of the proof on why a function cannot approach two different limits near $a$, so I will just list the relevant parts. If this is not enough/ambiguous ...
1
vote
3answers
201 views

Proof by Cases: $\operatorname{max}\{x,y\} + \operatorname{min}\{x,y\}=x+y$

So I'm told to "[u]se proof by cases to prove that $\operatorname{max}\{x,y\} + \operatorname{min}\{x,y\}=x+y$ for all real numbers $x$ and $y$." What does this mean?
0
votes
3answers
84 views

$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x\log \pi + (n-x)\log(1-\pi)\;\;?$

$$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x \log \pi + (n-x)\log(1-\pi)$$ this is what i have. i dont understand how $\binom{n}{x}$ disappears, but the rest is fine. I tried this, but it ...
1
vote
2answers
279 views

Strong Induction: Prove provided recurrence relation $a_n$ is odd.

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 ...
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
4answers
178 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
723 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
vote
2answers
63 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
648 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 ...
3
votes
4answers
1k views

Prove that if $x$ is odd then $x^2 -1$ is divisible by $8$.

If $x$ is odd then prove that $x^2-1$ is divisible by $8$. I start by writing: $x = 2k+1 $ where $k\in\mathbb{N}$. Then it follows that: $(2k+1)^2 -1 = 4k^2 +4k + 1 -1 $ Therefore: ...
0
votes
0answers
123 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
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: ...
1
vote
2answers
92 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
202 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
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 ...
4
votes
5answers
391 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: $a^2 < b^2$ ...