# Tagged Questions

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### 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 ...
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### 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 $n \geq 5$
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### 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.
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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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: ...
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### 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 ...
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### 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 $k>2$: $$p_k: 2^k>2k$$ ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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 ...
### 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: $a^2 < b^2$ ...