I'm trying to inductively prove:

$(a-b) | (a^n - b^n)$ where a,b are real numbers, and n is a real number and also 0

I've done my base case at zero, and made my inductive hypothesis:

$(a-b) | (a^{k+1} - b^{k-1}))$

I'm stuck here, a majority of the induction proofs I'm using involve proving something divides another, and would like some help on what to do with these. I have considered writing:

$(a^{k+1} - b^{k-1})) = s * (a-b)$, where $s$ is an element of the integers, but am still stuck there

thanks for answering!

  • 4
    $\begingroup$ I hope you know how to prove this without induction and that induction is called for only for induction practice: $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$. $\endgroup$ – Ethan Bolker Mar 23 '17 at 0:22
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    $\begingroup$ If $a,b$ are real numbers, I'm not even sure what that divisibility means. If I take $(a,b)=(4.7,2.1)$ then $a-b=2.6$, $a^3-b^3 =94.562$ and the ratio is clearly not integer. $\endgroup$ – Joffan Mar 23 '17 at 0:30

Hint (if sticking to induction): $a^{k+1}-b^{k+1}=a^{k+1}-ab^k + ab^k-b^{k+1}=a(a^k-b^k)+b^k(a-b)$

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