设数列{An}满足当n>1且n∈N*时,An=An-1/1+4An-1,且A1=1/5.(1)求证:数列{1/An}为等差数列

Question

设数列{An}满足当n>1且n∈N*时,An=An-1/1+4An-1,且A1=1/5.
(1)求证:数列{1/An}为等差数列

Answer 1

A(n) = A(n-1)/[1+4A(n-1)],
因A(1)=1/5,所以,A(n)不等于0,n=1,2,...
1/A(n) = [1 + 4A(n-1)]/A(n-1) = 1/A(n-1) + 4,
{1/A(n)}是首项为1/A(1) = 5,公差为4的等差数列.
1/A(n) = 5 + 4(n-1) = 4n + 1,
A(n) = 1/(4n+1),n = 1,2,...