Question

已知S_n=

1

1×2

+

1

2×3

+

1

3×4

+…+

1

n×(n+1)

（n∈N^*）的值是

2008

2009

，则n=______．

Answer 1

∵S_n=

1

1×2

+

1

2×3

+

1

3×4

+…+

1

n×(n+1)

=1-

1

2

+

1

2

-

1

3

+…+

1

n

-

1

n+1

=1-

1

n+1

=

n

n+1

∴

n

n+1

=

2008

2009

∴n=2008
故答案为：2008