问题 填空题
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=______.
答案

q3=

a4
a1
=27,求得q=3

∴an=3×3n-1=3n

∴bn=log3an=n,

1
bnbn+1
=
1
n(n+1)
=
1
n
-
1
n+1

∴Sn=1-

1
2
+
1
2
-
1
3
+…+
1
n
-
1
n+1
=1-
1
n+1
=
n
n+1

故答案为:

n
n+1

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