Mathematical Induction Problem

[Tracker]

The questions asks to use mathematical induction to prove the truth of the follow for [math] n \geq 1 [/math]

 

[math] a^n - b^n [/math] is divisible by [math] a - b [/math] for any integers a,b with [math] a-b \neq 0 [/math]

 

Proof Attempt:

 

[math]

 

n = 1[/math]

 

 

[math] a^n - b^n = a - b [/math]

 

[math]\frac{a - b}{a - b} = 1 [/math]

 

Now suppose that [math] n = k [/math] and the statement is true for [math] n = k [/math]

 

We must show that the statement is true for [math]n = k + 1[/math]

 

I tried using long division as it was suggested as a hint to consider the remainder:

 

[math] a^{k+1} - b^{k+1}[/math] divided by [math] a - b [/math] gives a remainder of [math] -b^{k+1} + ba^k [/math]

 

Any suggestions? Is this a historical proof by chance so I can see how it was done?

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Merged post follows:

Update of attempt

Using the algebraic expansion:

[math] a^n - b^n = (a - b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1} [/math] but for n = k + 1 solved the problem.

Edited March 24, 2010 by Tracker