题目

已知x+x分之1=2 求x^3+x^3分之一和对任意正整数n,猜想x^n+x^n分之一的值

提问时间：2021-01-21

答案

x^3 + 1/x^3
= (x + 1/x)^3 - 3 x^2/x - 3 x/x^2
= (x + 1/x)^3 - 3 (x + 3 /x)
= 2^3 - 3×2
=2
x^2 + 1/x^2
= (x + 1/x)^2 - 2 x 1/x
= 2^2 -2 = 2
x^n + 1/x^n
= (x^n + 1/x^n) (x + 1/x) / (x + 1/x)
= {x^(n+1) + 1/x^(n+1) + x^(n-1) + 1/x^(n-1)} / 2
得：x^(n+1) + 1/x^(n+1) = 2 {x^n + 1/x^n} - { x^(n-1) + 1/x^(n-1)}
由上式可知：
x^4 + 1/x^4 = 2 (x^3 + 1/x^3) - (x^2 + 1/x^2) = 2×2 - 2 = 2
x^5 + 1/x^5 = 2 (x^4 + 1/x^4) - (x^3 + 1/x^3) = 2×2 - 2 = 2
.
x^n + 1/x^n = 2