已知等比数列{an}中,a1=3,a4=81,若数列{bn}满足bn=log3an,则数列{1/bnbn+1}的前n项和Sn=_.

问题描述:

已知等比数列{an}中,a1=3,a4=81,若数列{bn}满足bn=log3an,则数列{

1
bnbn+1
}的前n项和Sn=______.

设公比为q,则a4=a1q3=3q3=81,解得q=3,所以an=3×3n-1=3n,bn=log3an=log33n=n,所以1bnbn+1=1n(n+1)=1n−1n+1,Sn=1b1b2+1b2b3+…+1bnbn+1=1-12+12−13+…+1n−1n+1=1-1n+1=nn+1,故答案为:nn+1....