已知数列{an}满足a1=a,an=1/[a(n-1)]+1(n>=2),若a4=0,则a等于?

题目

已知数列{an}满足a1=a,an=1/[a(n-1)]+1(n>=2),若a4=0,则a等于?

答案

由已知有a2=1/a1+1=1/a+1=(a+1)/a,a3=1/a2+1=a/(a+1)+1=(2a+1)/(a+1)
a4=1/a3+1=(a+1)/(2a+1)+1=(3a+2)/(2a+1)=0,解得a=-2/3