化简1/6n(n+1)(2n+1)

题目

化简1/6n(n+1)(2n+1)
1/6是六分之一

答案

显然
1/6n(n+1)(2n+1)
=1/6 *[1/n -1/(n+1)] * 1/(2n+1)
=1/6 * 1/n * 1/(2n+1) -1/6 *1/(n+1) * 1/(2n+1)
而
1/n * 1/(2n+1)
=2/2n * 1/(2n+1)
=2 * [1/2n -1/(2n+1)]
同理
1/(n+1) * 1/(2n+1)
=2/(2n+2) * 1/(2n+1)
=2*[1/(2n+1) -1/(2n+2)]
所以
1/6n(n+1)(2n+1)
=1/6 * 1/n * 1/(2n+1) -1/6 *1/(n+1) * 1/(2n+1)
=1/3 *[1/2n -1/(2n+1)] - 1/3 *[1/(2n+1) -1/(2n+2)]
=1/6n - 2/(6n+3) +1/(6n+6)