高三複數題

圖片參考：http://c.hiphotos.baidu.com/zhidao/wh%3D450%2C600/sign=db3303505266d0167e4c962ca21bf831/95eef01f3a292df502162c88bc315c6035a87352.jpg

(1 + i)²ⁿ / (1 - i) + (1 - i)²ⁿ / (1 + i) = 2ⁿ (1 + i)²ⁿ⁺¹ / ((1 - i)(1 + i)) + (1 - i)²ⁿ⁺¹ / ((1 + i)(1 - i)) = 2ⁿ(1 + i)²ⁿ⁺¹ + (1 - i)²ⁿ⁺¹ = 2ⁿ⁺¹( √2 (cos π/4 + i sin π/4) )²ⁿ⁺¹ + ( √2 (cos -π/4 + i sin -π/4) )²ⁿ⁺¹ = 2ⁿ⁺¹2ⁿ √2 ( cos (2n+1)π/4 + i sin (2n+1)π/4 + cos -(2n+1)π/4 + i sin -(2n+1)π/4 ) = 2ⁿ⁺¹ 2ⁿ √2 ( 2 cos (2n+1)π/4 ) = 2ⁿ⁺¹cos (2n+1)π/4 = √2 / 2(2n+1)π/4 = π/4 , 7π/4 , ....
當 (2n+1)π/4 = π/4
2n+1 = 1
n = 0 非正整數。
當 (2n+1)π/4 = 7π/4
2n+1 = 7
最小正整數 n = 3。