设x+1/x=—1,求x^3n+1+1/x^3n+1老师们帮帮我吧,在线等啦,快的!原题中文翻译一下：设x加上x分之一等于负一,求x的3n+1次方加上x的3n+1次方分之一.谢了~~~

Question

设x+1/x=—1,求x^3n+1+1/x^3n+1 老师们帮帮我吧,在线等啦,快的!
原题中文翻译一下：设x加上x分之一等于负一,求x的3n+1次方加上x的3n+1次方分之一.谢了~~~

Answer 1

递推 x+1/x=-1 x^2+1/x^2=(x+1/x)^2-2=-1 x^3+1/x^3=(x+1/x)(x^2+1/x^2)-(x+1/x)=2
一般x^(n+1)+1/x^(n+1)=(x+1/x)(x^n+1/x^n)-[x^(n-1)+1/x^(n-1)]=-(x^n+1/x^n)-[x^(n-1)+1/x^(n-1)]
按规律呢 x^4+1/x^4=-2-(-1)=-1 x^5+1/x^5=1-2=-1 x^6+1/x^6=1-(-1)=2,开始循环了
可知 x^(3n+1)+1/x^(3n+1)=-1 (*)
x^(3n+2)+1/x^(3n+2)=-1
x^(3n+3)+1/x^(3n+3)=2