设等比数列{an}的前n项和为Sn,已知a1=2011,且an+2an+1+an+2=0(n∈N*),则S2012=(  ) A.2011 B.2012 C.1 D.0

问题描述:

设等比数列{an}的前n项和为Sn,已知a1=2011,且an+2an+1+an+2=0(n∈N*),则S2012=(  )
A. 2011
B. 2012
C. 1
D. 0

设等比数列{an}的公比为q,∵an+2an+1+an+2=0,
∴an+2anq+anq2=0,即1+2q+q2=0,∴q=-1,
∴S2012=

a1(1−2012)
1−q
=
2011(1−1)
2
=0,
故选D.