Mathematical Induction with an Inequality

[yellowsnow]

Homework Statement

Prove that (n + 1)^{n - 1} < nⁿ for n ∈ Z⁺. [Hint: Induction is suggested. Write out the induction statement explicitly. Make one side of the inequality look like your induction hypothesis.]

Homework Equations

The Attempt at a Solution

^ That's what I have so far. I'm good with induction, for the most part, but not really for inequalities.

I tried to follow the hint, but I'm not sure if I did that right.

If anyone can help me out I'd really appreciate it.

Thanks!

 

[Mark44]

Your image is way too big (1996 pixels X 1869 pixels). Please shrink your image to about 800 x 600. Better yet, write your inequalities right in the text entry window.

 

[yellowsnow]

Your image is way too big (1996 pixels X 1869 pixels). Please shrink your image to about 800 x 600. Better yet, write your inequalities right in the text entry window.

Yeah, I resized it soon after posting (shows 613 x 573 now)

I'll try typing it up though :)

 

[Mark44]

You don't include it, but you also need to show that the statement is true in a base case, such as when n = 1. This is easy to show, since 2⁰ <= 1¹.

You have for the case n = k + 1, (k + 2)^k <= (k + 1)^{k + 1}. You seem to be assuming that this is true. Instead you need to show that it is true, using the statement in the induction hypothesis (i.e., (k + 1)^{k - 1} <= k^k).

I don't have any other advice or tips right now, but I'll give it some thought.

 

[Mark44]

I haven't taken this all the way through, so can't guarantee this is the way to go.

You need to show that (k + 2)^k <= (k + 1)^{k + 1}

Working with the left side, we have
(k + 2)^k = (k + 2)(k + 2)^{k - 1} = (k + 2) ( (k + 1) + 1)^{k -1}

Now expand the last factor using the Binomial Theorem, which says that
$$(a + b)^n = {{n}\choose{0}}a^nb^0 + {{n}\choose{1}}a^{n - 1}b^1 + {{n}\choose{2}}a^{n - 2}b^2 + ... + {{n}\choose{n}}a^{0}b^n$$