已知数列{an}满足a1=1,a2=3,an+2=3an+1-2an(n∈N*). (Ⅰ)证明:数列{an+1-an}是等比数列; (Ⅱ)求数列{an}的通项公式.

问题描述:

已知数列{an}满足a1=1,a2=3,an+2=3an+1-2an(n∈N*).
(Ⅰ)证明:数列{an+1-an}是等比数列;
(Ⅱ)求数列{an}的通项公式.

证明:(Ⅰ)∵an+2=3an+1-2an,∴an+2-an+1=2(an+1-an),∴an+2−an+1an+1−an=2(n∈N*).∵a1=1,a2=3,∴数列{an+1-an}是以a2-a1=2为首项,2为公比的等比数列.(Ⅱ)由(Ⅰ)得an+1-an=2n(n∈N*),∴an=(...