Question

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

Answer 1

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