Question

观察下列等式：a1＝11×2×3+12＝21×3；a2＝12×3×4+13＝32×4；a3＝13×4×5+14＝43×5…（1）猜想并

观察下列等式：a1＝11×2×3+12＝21×3；a2＝12×3×4+13＝32×4；a3＝13×4×5+14＝43×5…（1）猜想并写出第n个等式；（2）证明你猜想的正确性．

Answer 1

（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)

．