数列{an}前n项和为Sn,且Sn=n-5an-85,证明{an-1}是等比数列

Question

数列{an}前n项和为Sn,且Sn=n-5an-85,证明{an-1}是等比数列

Answer 1

Sn=n-5an-85
S1=1-5a1-85
即a1=1-5a1-85
解得a1=-14
an=Sn-S(n-1)
=n-5an-85-[(n-1)-5a(n-1)-85]
=-5an+5a(n-1)+1
6an=5a(n-1)+1
an=5a(n-1)/6+1/6
an-1=(5/6)[a(n-1)-1]
(an-1)/[a(n-1)-1=5/6
所以{an-1}是以an-1=-14-1=-15为首相q=5/6为公比的等比数列
an-1=(-15)*(5/6)^(n-1)