Question

已知數(shù)列{a_n}，滿足a1=1，

1

an+1

=

1

an

+1，S_n是數(shù)列{a_na_n+1}的前n項和，則S₂₀₁₁=______．

Answer 1

∵a1=1，

1

an+1

=

1

an

+1，
∴{

1

an

}是以1為首項以1為公差的等差數(shù)列
根據(jù)等差數(shù)列的通項公式可得，

1

an

=n即an=

1

n

∴anan+1=

1

n(n+1)

=

1

n

-

1

n+1

∴S2011=1-

1

2

+

1

2

-

1

3

+…+

1

2011

-

1

2012

=1-

1

2012

=

2011

2012

故答案為：

2011

2012