Question

观察下列等式：a1=

1

1×2×3

+

1

2

=

2

1×3

；a2=

1

2×3×4

+

1

3

=

3

2×4

；a3=

1

3×4×5

+

1

4

=

4

3×5

…
（1）猜想并写出第n个等式；
（2）证明你猜想的正确性．

Answer 1

分析：（1）通过观察得出规律后即可写出第n个等式；
（2）先对分式的左边通分，再相加后即可证出猜想的正确性．

解答：解：（1）an=

1

n(n+1)(n+2)

+

1

n+1

=

n+1

n(n+2)

 
（2）证明：左边=

1 n(n+1)(n+2)

+ n(n+2) n(n+1)(n+2)

=

1+n(n+2) n(n+1)(n+2)

= n2+2n+1 n(n+1)(n+2) = (n+1)2 n(n+1)(n+2) = n+1 n(n+2)

=右边
∴

1

n(n+1)(n+2)

+

1

n+1

=

n+1

n(n+2)

．

点评：此题考查了分式的加减，关键是能通过观察找出规律，并用式子表示出来．