Where is this question from? The factoring is incredibly complicated and the question itself needs more info to be answerable.
Here's the factoring:
(a^n + b^n)/(a^n-1 + b^n-1) = (a+b)/2
2(a^n + b^n) = (a+b)(a^n-1 + b^n-1)
2(a^n) + 2(b^n) = a(a^n-1) + b(b^n-1) + a(b^n-1) + b(a^n-1)
2(a^n) + 2(b^n) = a^n + b^n + a(b^n-1) + b(a^n-1)
2(a^n) - a^n + 2(b^n) - b^n = a(b^n-1) + b(a^n-1)
a^n + b^n = a(b^n-1) + b(a^n-1)
a^n - b(a^n-1) = a(b^n-1) - b^n
a^n-1(a-b) = b^n-1(a-b)
a^n-1 = b^n-1
Which is only true for ALL a and b if n - 1 = 0 or n = 1.
However, the question doesn't mention all a and b or tell us that a doesn't equal b. So, if we chose a=3 and b=3, then the two sides would be equal for all values of n. If we chose a=3 and b=-3, then the two sides would be equal for all odd values of n.
If the question told us that |a| != |b|, then 1 would be the correct answer. In future, please post all the answer choices.
I will get back on this sooon and yes nice question ! it was fun doing it 
