n∑k=0(nk)xkyn−k=(x+y)n since:
(1) For n=0,n∑k=0(nk)xkyn−k=(00)x0y0=1=(x+y)0
(2) Assume that n∑k=0(nk)xkyn−k=(x+y)n
(3) (x+y)(x+y)n= =(x+y)(xn+(n1)xn−1y+(n2)xn−2y2+⋯+(nk)xn−kyk+ +…(nn−1)xyn−1+yn)
=xn+1+[1+(n1)]xny+[(n1)+(n2)]xn−1y2+… [(nk−1)+(nk)]xn−k+1yk+⋯+[(nn−1)+1]xyn+yn+1
(4) Using Pascal's Rule (nr−1)+(nr)=(n+1r) and 1=(n0)=(nn) so that:
xn+1+[1+(n1)]xny+[(n1)+(n2)]xn−1y2+… [(nk−1)+(nk)]xn−k+1yk+⋯+[(nn−1)+1]xyn+yn+1=
(n+10)xn+1+(n+11)xny+(n+12)xn−1y2+… (n+1k)xn−k+1yk+⋯+(n+1n−1)xyn+(n+1n+1)yn+1=n+1∑k=0(n+1k)xkyn−k
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