Question

数列{an}的前n项和记为Sn,已知a1=1，an+1=n+2/n*Sn（n=1，2，3…..）

证明：（1）数列{Sn/n}是等比数列
（2）Sn+|=4an

Answer 1

a(n+1)=[(n+2)/n]*Sn
n(S(n+1)-Sn) = (n+2)Sn
nS(n+1) = 2(n+1)Sn
S(n+1)/裂猜郑(n+1) = 2(Sn/兆亩n)
=>肆颂{Sn/n} 是等比数列, q=2

Sn/n = 2^(n-1) ( S1)
Sn = n.2^(n-1)
for n>=2
an = Sn -S(n-1)
=n.2^(n-1) - (n-1).2^(n-2)
=(n+1).2^(n-2)
4an =4(n+1).2^(n-2)
= (n+1).2^n
=S(n+1)