Question

观察下列各式,1/6=1/2*3=1/2-1/3;1/12=1/3*4=1/3-1/4:1/20=1/4*5=1/4-1

Answer 1

公式：

1/n(n+1) = 1/n - 1/(n+1),其中：n为正整数

证明：

1/n(n+1)

=[(n+1) - n] / n(n+1)

=(n+1) / n(n+1) - n / n(n+1)

=1/n - 1/(n+1)

(2)

1/(x-2)(x-3) = [(x-2)-(x-3)] / (x-2)(x-3) = (x-2)/(x-2)(x-3) - (x-3)/(x-2)(x-3)

=1(x-3) - 1/(x-2)

-2/(x-1)(x-3) =-[(x-1)-(x-3)] / (x-1)(x-3) = (x-2) (x-2)(x-3) - (x-3) (x-1)(x-3)

= -1/(x-3) + 1/(x-1)

1/(x-1)(x-2) =[(x-1)-(x-2)]/(x-1)(x-2) = (x-1)/(x-1)(x-2) - (x-2)/(x-1)(x-2)

=1/(x-2) - 1/(x-1)

上述各式相加：

左边=1/(x-3) - 1/(x-2) - 1/(x-3) + 1/(x-1) + 1/(x-2) - 1/(x-1)

= 0