1. $sumlimits_{i=1}^{n} i^2 = (2 imes n^3+3 imes n^2+n)/6$
证明一：
设$ f_n = sumlimits_{i=1}^{n} i^2 $,$ S_n = sumlimits_{i=1}^{n} i^3 $
$S_{n+1} = S_{n}+(i+1)^3 $
$S_{n+1} = sumlimits_{i=0}^{n} (i+1)^3 $
$S_{n+1} = sumlimits_{i=0}^{n} i^3 + 3 imes i^2 + 3 imes i +1$
$(n+1)^3 = sumlimits_{i=0}^{n} 3 imes i^2 + 3 imes i +1$
$n^3 +3 imes n^2 + 3 imes n + 1 = 3 f_n + 3 imes n imes (n+1)/2 +n+1$
整理一下就行了。
证明二：
利用$sumlimits_{i=1}^n C_i^k = C_{n+1}^{k+1}$(证明: $sumlimits_{i=1}^n C_i^k = C_1^{k+1} + sumlimits_{i=1}^n C_i^k = C_2^{k+1}+ sumlimits_{i=2}^n C_i^k = C_{n+1}^{k+1}$)
那么$sumlimits_{i=1}^n C_i^2 = C_{n+1}^3$,
然后得$sumlimits_{i=1}^n i imes (i-1) /2 = (n+1) imes n imes(n-1) /6$
$sumlimits_{i=1}^n i imes i - sumlimits_{i=1}^n i = sumlimits_{i=1}^n i imes i - n imes (n+1) /2$
也就是$f_n -n imes (n+1) /2 = (n+1) imes n imes (n-1) /3$
2.括号序列合法实质：对于任意前缀和都非负，交换实质：改变区间前缀和（+2/-2）