$\frac{1}{2}P(2,1)+\frac{2}{3}P(3,2)+\frac{3}{4}(4, 3)+...+\frac{n}{n+1}P(n+1,n)=(n+1)!-1$
2. Is $\displaystyle P(n,k)=\frac{n!}{(n-k)!}$? Then yes, the statement can be easily proved by induction.