蓝桥杯-构造(数学公式1/n = 1/(n+1) + 1/(n+1)n )

0.题目

1.题解

1.0 找规律

n = 1, 1/1 = 1/2 + 1/3 + 1/6
n = 2, 1/2 = 1/4 + 1/6 + 1/12
n = 3, 1/3 = 1/6 + 1/9 + 1/18
....
实际上, 1/6 = 1/12 + 1/12, 1/12 = 1/36 + 2/36 = 1/36 + 1/18
即 1/6 = 1/(62) + 1/(623/2) + 1/(623), 即2,3,6三种

1.1 构造

我们想要知道 1/n = 三个数相加,
而已经 有1/n = 1/(n+1) + 1/(n+1)n
又有 1/(n+1) = 1/(n+2) + 1/(n+2)(n+1)
即 1/n = 1/(n+1)n + 1/(n+2) + 1/(n+2)(n+1)
更多的次数同理向后迭代1/(n+2)...即可