Question

已知数列an的前n项和为sn且an+sn=n,cn=an-1,求证：数列cn是等比数列

Answer 1

an+Sn=n

Sn=n-an

n=1时,a1=S1=1-a1

a1=1/2

n≥2时,an=Sn-S(n-1)=n-an-[(n-1)-a(n-1)]

2an=a(n-1) +1

2an -2=a(n-1)-1

(an -2)/[a(n-1)-1]=1/2,为定值

a1-1=1/2-1=-1/2,数列{an -1}是以-1/2为首项,1/2为公比的等比数列

cn=an -1,数列{cn}是以-1/2为首项,1/2为公比的等比数列