Question

已知数列an,Sn是其前n项和,且满足3an=2Sn+n(n为正整数）,求证[an+1/2]为等比数列

Answer 1

3an=2Sn+n

3an+1=2Sn+1 +(n+1)

得:an+1=3an +1

设 an+1 +k =3(an+k)

得 :k=1/2

所以：(an+1 + 1/2)/3(an+ 1/2)=3

a1=S1

3a1=2a1 +1

a1=1

a2=3a1 +1=4

[a2 +1/2]/[a1+1/2]=3

所以 [an+1/2]为等比数列