Question

已知x+x分之1=2 求x^3+x^3分之一

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

Answer 2

x+1/x=2经过通分得（x²+1）/x=2， 移项得x²+1-2x=0，所以（x-1）²=0，所以x=1.
众所周知，1的多少次方都是1，所以结果可想而知

Answer 3

x+1/x = 2
(x+1/x)^2 = 2^2
x^2+1/x^2+2=4
x^2+1/x^2=2

x^3+1/x^3=(x+1/x)(x^2-x*1/x+1/x^2)
= (x+1/x) { (x^2+1/x^2)-1 }
= (x+1/x) { (x^2+1/x^2)-1 }
= 2/(2-1)
=2

Answer 4

(x+1/x)^3=2^3
x^3+3x +3/x+1/x^3=8
x^3+1/x^3+3(x +1/x)=8
x^3+1/x^3+3*2=8
x^3+1/x^3=8-6=2