已知公比为3的等比数列{bn}与数列{an}满足{bn}=3an,n∈N*,且a1=1. (1)判断{an}是何种数列,并给出证明; (2)若cn=1/anan+1,求数列{cn}的前n项和.

问题描述:

已知公比为3的等比数列{bn}与数列{an}满足{bn}=3an,n∈N*,且a1=1.
(1)判断{an}是何种数列,并给出证明;
(2)若cn=

1
anan+1
,求数列{cn}的前n项和.

(1)∵等比数列{bn}的公比为3∴bn+1bn=3an+13an=3an+1−an=3∴an+1-an=1∴{an}是等差数列(2)∵a1=1,an+1-an=1∴an=n则cn=1anan+1=1n(n+1)=1n-1n+1∴Sn=c1+c2+c3+…cn=(1-12)+(12-13)+(13-14)+…+(1n-1n+1...