Question

数列｛an｝的前n项和Sn,且Sn=n-5an-85.

Answer 1

1.Sn=n-5an-85

Sn-1=n-1-5a(n-1)-85

an=Sn-Sn-1=1-5an+5a(n-1)

则 6an=5a(n-1)+1

∴ 6an-6=5a(n-1)-5

即 (an-1)/[a(n-1)-1]=5/6

所以,数列{an-1}是以5/6为公比的等比数列

2.a1=S1=1-5a1-85 得 a1=-14

∵数列{an-1}是以a1-1=-15为首项,5/6为公比的等比数列

∴an-1=-15×(5/6)^(n-1)

故 an=1-15×(5/6)^(n-1)

Sn=n-5an-85

=n-5+75×(5/6)^(n-1)-85

=n+75×(5/6)^(n-1)-90

=n+90×(5/6)^n-90