数列{An}满足Sn=n^2An,若A1=1005,求An的表达式?
问题描述：
数列{An}满足Sn=n^2An,若A1=1005,求An的表达式?
答
An=(n-1)*An-1/(n+1)
答
S2=2^2A2=A2+A1
A2(2^2-1)=A1
A2=A1/(2^2-1)
S3=3^2A2=A3+S2=A3+2^2A2
A3=2^2A2/(3^2-1)=A12^2/(3^2-1)(2^2-1)
An=(n-1)^2An-1/(n^2-1)
=1005*1^2*2^2*3^2*...(n-1)^2/(n^2-1)...(2^2-1)
=1005*[1*2*3*...*(n-1)^2]/[(n^2-1)*...(2^2-)]