数列{xn}满足xn+1=1+2/xn(n∈R),x1=11/7求证：数列{1/(xn-2)+1/3}是等比数列,并求出xn的表达式

Question

数列{xn}满足xn+1=1+2/xn(n∈R),x1=11/7
求证：数列{1/(xn-2)+1/3}是等比数列,并求出xn的表达式

Answer 1

记{an}={1/(xn-2)+1/3}
通分化简：an = 1/(xn - 2) + 1/3 = (xn + 1)/3(xn - 2)
将 xn+1 = 1 + 2/xn 代入 an+1 = 1/(xn+1 - 2) + 1/3,：
an+1 = 1/(xn+1 - 2) + 1/3 = 1/[1 + 2/xn - 2] + 1/3
通分化简得：an+1 = 2(xn + 1)/3(2 - xn)
an+1/an = -2
因此{an}为等比数列,公比q = -2
首项 a1 = 1/(x1 - 2) + 1/3 = -2
{an}的通项公式：an = a1*q^(n-1)= (-2)^n
an=1/(xn - 2) + 1/3 = (-2)^n
解得 xn = 1/[(-2)^n - 1/3] + 2