已知数列{an}中a1=1,点(an,an+1)在函数y=3x+2的图象上(n∈N*). (I)证明:数列{an+1}是等比数列; (Ⅱ)求数列{an}的前n项和.

问题描述:

已知数列{an}中a1=1,点(anan+1)在函数y=3x+2的图象上(n∈N*)
(I)证明:数列{an+1}是等比数列;
(Ⅱ)求数列{an}的前n项和.

证明:(I)由题意可得,an+1=3an+2
则an+1+1=3(an+1)且a1+1=2
∴数列{an+1}是以2为首项,以3为公比的等比数列
(II)由(I)可得,an+1=2•3n−1
an=2•3n−1−1
Sn=(2•30−1)+(2•3−1)+…+(2•3n−1−1)
=2(1+3+…+3n-1)-n
=2

1−3n
1−3
−n=3n-1-n