Question

Sn=

1

1×2×3

+

1

2×3×4

+

1

3×4×5

+…+

1

n(n+1)(n+2)

，求S_n．

Answer 1

因为an=

1

n(n+1)(n+2)

=

1

2

[

1

n(n+1)

-

1

(n+1)(n+2)

]
所以Sn=

1

2

[

1

1×2

-

1

2×3

+

1

2×3

-

1

3×4

+…+

1

n(n+1)

-

1

(n+1)(n+2)

]
=

1

2

[

1

2

-

1

(n+1)(n+2)

]
=

n(n+3)

4(n+1)(n+2)