Question

已知数列an Sn是其前n项的和 a1=2 Sn+1=3Sn+n^2+2 设bn=an+n 证{bn}是等比数列

Answer 1

S(n+1)=3Sn+n^2+2

Sn=3S(n-1)+(n-1)^2+2

所以

a(n+1)=3an+2n-1

bn=an+n

b(n+1)=a(n+1)+(n+1)

所以

b(n+1)/bn=[a(n+1)+(n+1)]/(an+n)

=[3an+2n-1+n+1]/(an+n)

=3

所以是等比数列