A question on Induction

Question

Prove that: for any natural number $n\ge10$, $$2^n\ge n^3.$$ How can I prove by induction if $n\ge10$ and I must do that $2^n\ge n^3$?

I stop on this step: $$2^{n+1}\ge n^3+3n^2+3n+1.$$

Here's the question I'm trying to prove. I'm just not certain how I should approach the inductive / constructor step. I proved induction wrote $n+1$ instead of $n$.

I think I have everything right until the induction step. But I don’t know what I should do next step, because I had never solve inequality before I began ask questions on this site.

Welcome to MSE. It is in your best interest that you use MathJax. — José Carlos Santos, May 29 '18 at 06:19

score 0 · Answer 1 · answered May 29 '18 at 06:20

0

HINT:

The basis for induction is easy (just substitute $n=10$).

For the induction hypothesis, suppose that $n=k$ holds, so $$2^k\ge k^3$$ for some $k\ge 10$.

Consider $n=k+1$:

$$2^{k+1}=2\cdot2^k\ge2k^3$$ and all you have to do now is show that $$2k^3\ge(k+1)^3$$ for $k\ge10$.

answered May 29 '18 at 06:20

ə̷̶̸͇̘̜́̍͗̂̄︣͟

27,005

Thank you very much. But I don't understand why I should show that 2k3≥(k+1)3. You wrote above 2k+1=2⋅2k≥2k3. There I should prove that – K.Anika May 29 '18 at 06:30
2k3<=2k+1=2⋅2k. – K.Anika May 29 '18 at 06:31
No, you already know that $2^{k+1}=2\cdot2^k\ge2k^3$ from the induction hypothesis! So to show that $$2^{k+1}\ge(k+1)^3$$ you just have to prove the final polynomial inequality. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ May 29 '18 at 06:32
And how have I prove the final polynomial inequality? – K.Anika May 29 '18 at 06:34
Show some attempt first then I'll provide a further hint should you require one. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ May 29 '18 at 06:35
But I don't know how I should do first step. – K.Anika May 29 '18 at 06:38
Well do you know of the Binomial Theorem? – ə̷̶̸͇̘̜́̍͗̂̄︣͟ May 29 '18 at 06:39
No, I don't know. – K.Anika May 29 '18 at 06:40
OK, then expand $(k+1)^3=(k+1)(k+1)(k+1)$. Can you do that? – ə̷̶̸͇̘̜́̍͗̂̄︣͟ May 29 '18 at 06:41
Yes, I can do that . Thank you very much! I must to go. – K.Anika May 29 '18 at 06:43

score 0 · Answer 2 · answered May 29 '18 at 08:18

$2^(n+1) = 2^n*2 >= 2n^3$

Compare $2n^3$ to $n^3 + 3n^2 + 3n + 1$

$n^3 vs. 3n^2 + 3n + 1$

We can start to see that the left hand side is greater than the right hand side for n>=10. It is true at n=10.

Consider each side as two functions. Then taking derivatives we have $3n^2$ and $6n + 3$

The left function is increasing faster than the right function for n > 10.

A question on Induction

2 Answers2