Question

数列an的首项为1,前n项和是Sn,存在常数A,B使an+Sn=An+B对任意正整数n都成立.

Answer 1

Sn=an+An+B

Sn-1=a(n-1)+A(n-1)+B

an=Sn-Sn-1=an-a(n-1)+A

a(n-1)=A,为定值.

a1+a1=A+B

2A=A+B B=A

2a1=2A,a1=1代入得A=1

an=1

数列{an}为各项均等于1的常数数列,通项公式an=1