An=C(1,n)a1+C(2,n)a2+…C(n,n)an,若an=1+2+3+……+n(n∈N),试用n表示An._数学解答

An=C(1,n)a1+C(2,n)a2+…C(n,n)an,
若an=1+2+3+……+n(n∈N),试用n表示An.

人气：160 ℃　时间：2019-12-16 09:35:16

解答

C(k,n)ak=n!/((n-k)!*k!)*(k(k+1))/2
=(n-1)!/((n-k)!(k-1)!)*(n(k+1))/2
=C(k-1,n-1)*n/2*(k+1)
An=n/2*[C(0,n-1)*2+C(1,n-1)*3+……+C(n-1,n-1)*(n+1)]
=n/4*[C(0,n-1)*(n+3)+C(1,n-1)*(n+3)+……+C(n-1,n-1)*(n+3)] {逆序相加}
=n(n+3)/4*[C(0,n-1)+C(1,n-1)+……+C(n-1,n-1)]
=n(n+3)/4*2^(n-1)
=n(n+3)*2^(n-3)