Prove by induction divisibility question

[shaner243]

Homework Statement

Given that n is a positive integer, prove by induction that (a^n-b^n) is divisible by (a-b)

Homework Equations

n = k
n = k+1

The Attempt at a Solution

a^(k+1) - b(k+1) = (a-b)A, where A is a positive integer. I am lost from here or not sure if this is even the right attempt. Please help!

 

[MATLABdude]

Welcome to PhysicsForums!

I've requested that a mod move this thread to the math help section, rather than the Engineering / Comp Sci / Other section (not that we're not capable, but just for sake of completeness--that last phrase you'll probably hear a lot in your proofs class).

In general, proof by induction takes the following steps:

1. Show that it works for a trivial case, for instance k=0 or k=1
2. Assume that case n=k works
3. Based on this assumption, show that n=k+1 follows, usually by algebraically rearranging things to show something involving the n=k case.

I'll start you off on this third step. You'll need to factor the following a little (and yes, it's possible, you just need to add/subtract a little):
$$(a^{k+1} - b^{k+1})$$

HINT:
$$(a^2-b^2)(a+b) = a^3 + a^2b - ab^2 - b^3$$
$$\Rightarrow a^3 - b^3 = ?$$

EDIT: Error in hint fixed, courtesy of Mark44!

 

[Mark44]

Welcome to PhysicsForums!

I've requested that a mod move this thread to the math help section, rather than the Engineering / Comp Sci / Other section (not that we're not capable, but just for sake of completeness--that last phrase you'll probably hear a lot in your proofs class).

In general, proof by induction takes the following steps:

1. Show that it works for a trivial case, for instance k=0 or k=1
2. Assume that case n=k works
3. Based on this assumption, show that n=k+1 follows, usually by algebraically rearranging things to show something involving the n=k case.

I'll start you off on this third step. You'll need to factor the following a little (and yes, it's possible, you just need to add/subtract a little):
$$(a^{k+1} - b^{k+1})$$

HINT:
$$(a^2-b^2)(a+b) = (a^3 - a^2b + ab^2 - b^3)$$
$$(a^3 - b^3) = ?$$

I see where you're going, but there's a sign error in the first line above.
$$(a^2-b^2)(a+b) = a^3 - ab^2 + a^2b - b^3$$