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So I wanted to prove this statement through Induction and I started as normal,proving that this statement is true for two random values of $n,p \in \mathbb{N} \space with\space p\geq 3$.

In the second step though I encounter a problem continuing this proof. I tried to conclude that if this statement is true for $n$ then it's also true for $n+1$ but none of my ideas to get there seem to work.

I am also hesitant about it only requering this step, and not also proving it for $p+1$

Konstantinos Zafeiris
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  • @lulu I want to prove that the inequality is true for the given set of natural numbers, and no, no given assumption regarding n and p, and that's a problem I encountered too – Konstantinos Zafeiris Nov 01 '17 at 21:07
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    So, first, I think it suffices to prove this for $p=3$, since for all $p>3$, we have $p^n>3^n$. Also, to prove $3^n>n^2$, if you start the inductive proof by $n=2$, then the inductive step from $n$ to $n+1$ should be straightforward – A. M. Nov 01 '17 at 21:11
  • This could be useful https://math.stackexchange.com/questions/2489665/prove-that-if-k-in-mathbb-n-and-a1-then-lim-limits-n-to-infty-fra/ – rtybase Nov 01 '17 at 21:13
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    I have posted a non-inductive proof below. Of course, if the whole point was to prove the thing inductively, then this might not help much. – lulu Nov 01 '17 at 21:14

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If $p\geq 3$, then $p^n\geq 3^n$, for all $n\in\mathbb N$. Thus, it is enough to prove $3^n>n^2$. It is easy to verify that it holds for $n=1,2$.

Now, assume that the claim holds for $n\geq 2$. Then, we need to prove that

$$3^{n+1}> 3n^2 \geq (n+1)^2,$$

but, this easily follows from the fact that $2n^2-2n-1$ has roots $n_{1,2}=\frac{1\pm\sqrt 3}2$, so $3n^2\geq (n+1)^2$ for $n\geq 2 > \frac{1+\sqrt 3}2$.

Ennar
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Base case: $p^1 > 1^2$. That's true as $p \ge 3$

Induction case: Assume we know $p^k > k^2$

Then $p^{k+1} = p*p^k > pk^2 \ge 3nk^2$

$= k^2 + 2k^2 = k^2 + 2k*k = k^2 +2k(k-1) + 2k = k^2 + 2k + 2k(k-1)$.

Hmmm, if $2k(k-1) \ge 1$ we would be done as that would mean $k^2 + 2k + 2k(k-1)\ge k^2 + 2k + 1 = (k+1)^2$.

But our base case was $n=1 $ and $2*1(1-1) =0$. So we need a second base case.

If $n=2$ then $p^2 > 2^3$ because $p \ge 3 > 2$.

So for $n \ge 2$ $2n(n-1) \ge 2*2*(2-1) = 4 > 1$.

So back to our induction step:

If $p^k > k^2$ and $k \ge 2$ then

$p^{k+1} \ge 3p^k > 3k^2 = k^2 + 2k + 2k(k-1) > k^2 + 2k + 1 = (k+1)^2$.

...

So we are done.

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We dont have to prove $P(p) \implies P(p+1)$ because there's nothing about it being true for $p+1$ that requires we know it is true for $p$ first. It's merely that if $p \ge 3$ it is true-- just because $p \ge 3$. We don't need to know it is true for $p-1$ first.

Example: You don't need to inductively prove that if $p \ge 17$ then $p$ is positive . $p \ge 17 > 0$ and we have no reason to have to prove it for $p-1$ first.

fleablood
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Here is a non-inductive solution:

We consider the function $$f(x)=x^{2/x}$$ for $x≥1$

It is easy to differentiate this and thereby show that the unique maximum occurs at $x=e$. We then remark that $$f(e)\approx 2.087$$ In particular, $3>f(x)$ for all $x≥1$ and that suffices to prove your claim.

lulu
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  • That's a solution to the problem, and one that I prefer too, but it's through differential calculus. I am searching for an approach to this through induction. – Konstantinos Zafeiris Nov 01 '17 at 21:14
  • Fair enough. I'll leave this up, as it does solve the problem, but I'll edit it to take note of the objection. – lulu Nov 01 '17 at 21:15
  • We can write: $3^{n+1}=3.3^n>3n^2$. Now, we need to show this to be greater than $(n+1)^2=n^2+2n+1$, or equivalently $2n^2>2n+1$, or $n^2>n+0.5$, and this is true for $n\geq 2$. – A. M. Nov 01 '17 at 21:18
  • @A.M. That's exactly the point that I am stuck – Konstantinos Zafeiris Nov 01 '17 at 21:22