a1=2.S(n+1)=(n+1/n)Sn+2(n+1)^2.求an 的通项公式
问题描述:
a1=2.S(n+1)=(n+1/n)Sn+2(n+1)^2.求an 的通项公式
答
S(n+1)=[(n+1)/n]Sn+2(n+1)^2
nS(n+1)=(n+1)Sn+2n(n+1)^2
S(n+1)/(n+1) - Sn/n = 2(n+1)
Sn/n - S(n-1)/(n-1) = 2n
Sn/n - S1/1 = 2[ 2+3+...+n]
Sn/n = 2(1+2+...+n)
=n(n+1)
Sn = n^2.(n+1)
an = Sn -S(n-1)
=n^2.(n+1) - n(n-1)^2
= n[ n^2+n - n^2+2n-1]
=n(3n-1)
ie
an =n(3n-1)