已知等比数列{an}中,a1=3,a4=81,当数列{bn}满足bn=log3an,则数列{1bnbn+1}的前2013项和S2013为______.-高二数学

题目简介

已知等比数列{an}中,a1=3,a4=81,当数列{bn}满足bn=log3an,则数列{1bnbn+1}的前2013项和S2013为______.-高二数学

题目详情

已知等比数列{an}中,a1=3,a4=81,当数列{bn}满足bn=log3an,则数列{
1
bnbn+1
}
的前2013项和S2013为______.
题型:填空题难度:中档来源:不详

答案

设等比数列{an}的公比为q,∵a1=3,a4=81,∴81=3×q3,解得q=3.
an=3n
∴bn=log3an=log33n=n.
class="stub"1
bnbn+1
=class="stub"1
n(n+1)
=class="stub"1
n
-class="stub"1
n+1

∴Sn=(1-class="stub"1
2
)+(class="stub"1
2
-class="stub"1
3
)+
…+(class="stub"1
n
-class="stub"1
n+1
)
=1-class="stub"1
n+1
=class="stub"n
n+1

S2013=class="stub"2013
2014

故答案为class="stub"2013
2014

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