若数列an满足a1＝1/2.a1+a2+a3+……+an＝n^2an则数列an的前60项和为
问题描述：
若数列an满足a1＝1/2.a1+a2+a3+……+an＝n^2an
则数列an的前60项和为
答
an/a1=2/[n(n+1)]
an/(1/2)=2*[1/n-1/n(n+1)]
an=1/n-1/(n+1)
s60=1-1/2+1/2-1/3+........+1/60-1/61)
=1-1/61
=60/61
答
s(n)=n^2a(n)
a(n+1)=s(n+1)-s(n)=(n+1)^2a(n+1)-n^2a(n)
n(n+2)a(n+1)=n^2a(n)
(n+2)a(n+1)=na(n)
(n+2)(n+1)a(n+1)=(n+1)na(n)
(n+2)(n+1)a(n+1)=(n+1)na(n)=...=(1+1)*1*a(1)=1
a(n)=1/[n(n+1)] = 1/n - 1/(n+1)S60=60^2(1/60-1/61)=60/61
答
a1+a2+a3+……+an＝n^2ana1+a2+a3+……+a(n-1)＝(n-1)^2a(n-1)两式相减得an＝n^2an-(n-1)^2a(n-1)(n^2-1)an=(n-1)^2a(n-1)an/a(n-1)=(n-1)^2/(n^2-1)an/a(n-1)=(n-1)^2/[(n-1)(n+1)]an/a(n-1)=(n-1)/(n+1)an/a(n-1)=...