Question

按规律排列的一列数2,8,20,40,70,112,168.,则第n个数是?

Answer 1

相邻数作差可得：6 12 20 30 42 56

2*3 3*4 4*5 5*6 6*7 7*8

可知第n个数是 1*2 + 2*3 + 3*4 + ...+ n*(n+1)

= (1²+1) + (2²+2) + (3²+3) + ...+ (n²+n)

= (1²+2²+3²+...+n²) + (1+2+3+...+n)

= n(n+1)(2n+1)/6 + n(n+1)/2

= [n(n+1)/6] * (2n+1+3)

= n(n+1)(n+2)/3