Math plz help 已知a+1/a=1,求a^2015+1/a^2015的值.?

2015-10-20 11:59 am
更新1:

可以不用Cos嗎?

回答 (2)

2015-10-20 2:46 pm
✔ 最佳答案
因為 a + 1/a = 1
所以 a² + 2 + 1/a² = 1
即 a² + 1/a² = -1 ⋯⋯⋯⋯ ①
又 a + 1/a = 1
==> a² - a + 1 = 0
==> (a + 1)(a² - a + 1) = 0
==> a³ + 1 = 0
==> a³ = -1 ⋯⋯⋯⋯⋯⋯ ②
所以
a²⁰¹⁵ + 1/a²⁰¹⁵
= a²[(a³)⁶⁷¹] + 1/a²[(a³)⁶⁷¹]
= a²[(-1)⁶⁷¹] + 1/a²[(-1)⁶⁷¹] ⋯⋯ (從 ② 得知 a³ = -1)
= -a² - 1/a²
= -(a² + 1/a²)
= -(-1) ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯ (從 ① 得知 a² + 1/a² = -1)
= 1
2015-10-20 12:17 pm
Sol
a+1/a=1
a^2+1=a
a^2-a+1=0
(a+1)(a^2-a+1)=0
a^3+1=0
a^3=-1
a^2015=a^2013*a^2=-a^2
a^2015+1/a^2015
=-a^2-1/a^2
=(-a+1)/a^2
=-a^2/a^2
=-1


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