求證1P1 + 2*(2P2)+ 3*(3P3)+ .+n*(nPn)=(n+1)P(n+1)- 1

求證1P1 + 2*(2P2)+ 3*(3P3)+ .+n*(nPn)=(n+1)P(n+1)- 1

不難,用數學歸納法.證1.n=1時易證結論成立;2.設n=k時結論成立,即1*1!+2*2!+3*3!.k*k!=(k+1)!-1則n=k+1時1*1!+2*2!.+k*k!+(k+1)*(k+1)!=(k+2)!-1.兩式相减得:左邊=(K+1)*(k+1)!,右邊=(k+2)!-(k+1)!=(k+1)*(k+1)!.左式等於右式,即當n=k+1時結論也成立.有數學歸納法可知結論成立