Question

已知等比數(shù)列{a_n}中，a₁=3，a₄=81，當(dāng)數(shù)列{b_n}滿足b_n=log₃a_n，則數(shù)列{

1

bnbn+1

}的前2013項和S₂₀₁₃為______．

Answer 1

設(shè)等比數(shù)列{a_n}的公比為q，∵a₁=3，a₄=81，∴81=3×q³，解得q=3．
∴an=3n．
∴b_n=log₃a_n=log33n=n．
∴

1

bnbn+1

=

1

n(n+1)

=

1

n

-

1

n+1

．
∴S_n=(1-

1

2

)+(

1

2

-

1

3

)+…+(

1

n

-

1

n+1

)=1-

1

n+1

=

n

n+1

．
∴S2013=

2013

2014

．
故答案為

2013

2014

．