Question

已知数列{an}满足a1=1/2,a(n+1)*an-2a(n+1)+1=0

注：a后面的都为下标的。 1 求证：数列1/a(n-1)是等差数列2 求an

Answer 1

1.
应该是{1/(An-1)}是等差数列
A(n+1)×(An-2)=-1
A(n+1)=-1/(An-2)
A(n+1)-1=-1/(An-2)-1=-(An-1)/(An-2)
1/(A(n+1)-1)=-(An-2)/(An-1)=1/(An-1)-1
1/(A1-1)=1/(1/2-1)=-2
{1/(An-1)}是-2为首项，-1为公比的公差的等差数列

2.
1/(An-1)=-2-(n-1)=-n-1
An-1=-1/(n+1)
An=1-1/(n+1)=n/(n+1)