Question

show that a^n - b^n is divisible by a+ b when n is an even positive integer but not if n is odd​

Answer 1

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Answer 2

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If a=b,an=bn

or an−bn=(a−b)(an−1+an−2b+⋯+abn−2+bn−1)

If a=−b,a2m+1=(−b)2m+1=−b2m+1

Induction :

an−bn=a(an−1−bn−1)+bn−1(a−b)

a2m+1+b2m+1=a2(a2m−1+b2m−1)−b2m−1(a2−b2)

Step-by-step explanation:

order by

votes

Up vote

5

Down vote

If a=b,an=bn

or an−bn=(a−b)(an−1+an−2b+⋯+abn−2+bn−1)

If a=−b,a2m+1=(−b)2m+1=−b2m+1

Induction :

an−bn=a(an−1−bn−1)+bn−1(a−b)

a2m+1+b2m+1=a2(a2m−1+b2m−1)−b2m−1(a2−b2)