Question

数列{A(n)}满足：A(1)=1/2，A(n+1)*A(n)-2A(n+1)+1=0

求证：数列{1/(A(n)-1)}是等差数列；

Answer 1

A(n)-2+(1/An+1)=0
A(n-1)=1-1/(An+1)=0
A(n-1)=1-1/(An+1)=A(n+1)+1/A(n+1)-1
1/A(n-1)=A(n+1)/(A(n+1)-1=A(n-1)+1/A(n+1)
1/An-1=1+1/A(n+1)-1
1/(A(n+1)-1-1/A(n)-1=1