- 数学：数列问题18.已知数列{an},满足a(n+1)=(an+
- 18.已知数列{an},满足a(n+1)=(an+3)/(an-1)且a1=2.
(1)证明数列{(an+1)/(an-3)}(n∈N+)是等比数列.
(2)求出数列{an}的前n项和Sn.
- 证明:[a(n+1)+1]/[a(n+1)-3]=(an+3+an-1)/(an+3-3an+3)
=(2an+2)/(-2an+6)=-(an+1)/(an-3)
后一项等于前一项的相反数,所以{(an+1)/(an-3)}是首项为-3公比为-1的等比数列
(2)因为[4/(an-3)]+1=3*(-1)^n
所以(an-3)/4=(-1)^n/3
an=4(-1)^n/3+3
Sn=3n+2/3+(-1)^(n+1)