Question

（1+X）的3次方+（1+X）的4次方+。。。+（1+X）的N+2次方，的展开式中含X的2次方的项的系数是多少？

Answer 1

(1+X)^3+（1+X)^4+。。。+（1+X)^(n+2)
= (1+x)^3 [ 1+ (x+1) + (1+x)^2 .........+ (1+x)^(n-1)]
还记得等比级数吗﹖

= (1+x)^3[ ( 1-(1+x)^n )/ (1-(1+x)) ]
= (1+x)^3 [ (1-(1+x)^n )/(-x)]
= - (1+x)^3 /x (1-(1+x)^n )
= - (1+x)^3 /x (1-(1+nx + n(n-1)x^2+ n(n-1)(n-2)x^3 /6+.......)
= (1+ 3x+ 3x^2+ x^3 )/x ( nx + n(n-1)x^2+ n(n-1)(n-2)x^3 /6+.......)
= (1+ 3x+ 3x^2+ x^3 ) ( n + n(n-1)x+ n(n-1)(n-2)x^2 /6+.......)
可能形成x^2的项有 (1)n(n-1)(n-2)x^2 /6 + 3x n(n-1)x + n 3x^2
= (n^3/6 -3n^2/6+2n/6 + 3n^2-3n + 3n )x^2
= (n^3/6 -n^2/2+n/3 + 3n^2-3n + 3n )x^2
= (n^3/6 -n^2/2+n/3 + 3n^2-3n + 3n )x^2
= (n^3/6-5/2n^2+n/3)x^2
= (n^3-15n^2+2n)x^2/6
含X的2次方的项的系数是= (n^3-15n^2+2n)/6