数列{an}满足a1=-1,且an=3an1-2n+3(n=2,3~)(1)求a2,a3,并证明{an-n}是等比例数列;(2)求an

Question

数列{an}满足a1=-1,且an=3an_1-2n+3(n=2,3~)(1)求a2,a3,并证明{an-n}是等比例数列;(2)求an

Answer 1

(1)
∵a1=-1,且an=3an_1-2n+3
∴a2=3a1-2×2+3=-4
a3=3a2-2×3+3=-15
根据题意
a(n+1)=3an-2(n+1)+3=3an-2n+1
∴[a(n+1)-(n+1)]/(an-n)
=[3an-2n+1-n-1]/(an-n)
=[3an-3n]/(an-n)
=3
∴{an-n}是等比例数列,公比为3
（2）
由(1)知
an-n=(a1-1)*3^(n-1)
∴an=n-2*3^(n-1)