Question

请帮我推导一个式子

Answer 1

nCr = n!/(r!(n-r)!)
To prove:
(3C2)+(4C2)+..+ ((n+2)C2) = ((n+3)C3) - (2C2)
LS
=(3C2)+(4C2)+..+ ((n+2)C2)
=3!/(2!1!)+4!/(2!2!)+..+ (n+2)!/(2!n!)
=(1/2)[3/1!+4/2!+..+ (n+2)!/n! ]
= (1/2)[ 3(2) + 4(3)+ 5(4))+ .... + (n+2)(n+1) ]
= (1/2)[ 1.2+ 2.3 +3.4 +..+ (n+1)(n+2) ] - 1

consider
n(n+1) = (1/3)[ n(n+1)(n+3) - (n-1)n(n+1)]

summation(i:1->n+1) i(i+1)
= (1/3)(n+1)(n+2)(n+3)

LS
= (1/2)[ 1.2+ 2.3 +3.4 +..+ (n+1)(n+2) ] - 1
= (1/6)(n+1)(n+2)(n+3) - 1
= (n+3)!/(3!n!) -1
=((n+3)C2) - (2C2) = RS

追问

太深奥了

Answer 2

这个需要一个引理：Cn取3=Cn-1取2+Cn-1取3（证明只要用定义全部展开即可）
于是移项可得 Cn+2取2=Cn+3取3-Cn+2取3
把上式每一项展开，就相当于裂项相消了，结果很明显了，楼主自己列式子试一试吧

Answer 3

建议用数学归纳法证~