prove identity

you may expand it from RHS

(a + b)(a^(n-1) - a^(n-2) b + a^(n-3) b^2 - … - a b^(n-2) + b^(n-1))

= a(a^(n-1) - a^(n-2) b + a^(n-3) b^2 - … - a b^(n-2) + b^(n-1))
+ b(a^(n-1) - a^(n-2) b + a^(n-3) b^2 - … - a b^(n-2) + b^(n-1))

= a^n - a^(n-1) b + a^(n-2) b^2 - … - a^2 b^(n-2) + ab^(n-1)
+ a^(n-1) b - a^(n-2) b^2 + a^(n-3) b^3 - … - a b^(n-1) + b^n

= a^n + b^n

thxthxthx.

其實 for odd n,

aⁿ + bⁿ 抽出 (a + b) 之後，餘下的會一加一減～

至於 for even n,

aⁿ - bⁿ 其實可抽 (a + b) 也可抽 (a - b)
抽 (a + b) 餘下的會一加一減～
抽 (a - b) 餘下的全部加～

2014-05-06 19:27:50 補充：
請看三個例子：

http://www.wolframalpha.com/input/?i=(a%5E7%2Bb%5E7)%2F(a%2Bb)&t=crmtb01

http://www.wolframalpha.com/input/?i=%28a%5E8-b%5E8%29%2F%28a%2Bb%29

http://www.wolframalpha.com/input/?i=%28a%5E8-b%5E8%29%2F%28a-b%29