1. ## Math Induction Problem

Hello,

I have a mathematical induction problem I need some assistance on.

Here's the problem:

Prove by induction on n that, n^3 (is less than or equal to) 2^n for n (greater than or equal to) 10 a positive integer.

So far, I can only get part of the problem (if I am correct to that point?) then I get stuck. Any help would be greatly appreciated. Thanks in advance!

CaptainBlack: It is n to the 3rd power less than or equal to 2 to the power of n for n greater than or equal to 10. I think you might of read it wrong.

I know the first step is to substiute k so you get k^3 <= 2^K
Then step two substitute K + 1 to get (k + 1)^3 <= 2^(k + 1)

From here I have trouble.

Thanks again,

2. Originally Posted by MathStudent1
Hello,

I have a mathematical induction problem I need some assistance on.

Here's the problem:

Prove by induction on n that, n^3 (is less than or equal to) 2^n for n (greater than or equal to) 10 a positive integer.

So far, I can only get part of the problem (if I am correct to that point?) then I get stuck. Any help would be greatly appreciated. Thanks in advance!
For $n \ge 1$, $n^3 \ge n^2$ not $n^3 \le n^2$, so its no supprise you are having trouble proving what you say you are trying to prove.

RonL

RonL

3. Well I can have a go. If anyone posts after me, listen to them and not me though. I am not 100% sure I remember the method.

To prove by induction you first have to show it is true for some value. So put in the number 10 and show that the inequality holds.

Then you have to assume
{1} k³ (less or equal to) 2^k

Then you must show that this is also true for k+1

(k+1)³ = k³ + 3k² + 3k + 1 [less than or equal to] 2k³ (for large enough k)
2k³ [less than or equal to] 2 x 2^k (by {1})
2 x 2^k = 2^k+1

Which shows it is true for k+1