a1=3,an=Sn-1+2n次方.n大于或等于2.求an和Sn

Question

a1=3,an=Sn-1+2n次方.n大于或等于2.求an和Sn

Answer 1

为表达清析,用An表示an
A1=3
A2=11
An=Sn-1+2n^2.1
An+1=Sn+2(n+1)^2.2
2式-1式
An+1-An=An+2(n+1)
An+1=2An+2n(n+1)
An+1+4(n+1)+6=2(An+4n+6).3
令Bn=(An+4n+6)
3式变为
Bn+1=2Bn (n≥2) B2=A2+4*2+6=25
Bn=25*2^(n-2) (n≥2)
即：An+4n+6=25*2^(n-2)
An=25*2^(n-2)-4n-6 (n≥2)
Sn=25*2^(n-1)-2n(n+4)-12
数列必杀大王!