急,我有d關於A Maths o既問題想問吓大家 (MI)2

命題 P(n): 1x1! + 2x2! + ... + nxn! = (n+1)! - 1
其中 n = 1, 2, 3, ..... (正整數)

當 n = 1:
左方 = 1x1! = 1
右方 = (1+1)! - 1 = 1
左方 = 右方
因此，P(n)正確。

設 P(k) 正確，所以
1x1! + 2x2! + ... + kxk! = (k+1)! - 1

當 n = k+1
求證：1x1! + 2x2! + ... + kxk! + (k+1)x(k+1)! = (K+2)! - 1
證明：
左方
= (1x1! + 2x2! + ... + kxk!) + (k+1)x(k+1)!
= [(k+1)! - 1] + (k+1)x(k+1)!
= (k+1)! + (k+1)x(k+1)! - 1
= [1 + (k+1)]x(k+1)! - 1
= [k+2]x(k+1)! - 1
= (k+2)! - 1
= 右方

根據數學歸納法的原理，當 n 為正整數時﹐P(n) 正確。
=

For n = k +1.
1*1! + .... + (k+1)*(k+1)! = (k +1)! - 1 + (k+1)(k+1)!
= (k+1)!(1 + k+1) - 1
=(k+1)!(k+2) - 1
= (k + 2)! - 1
= [(k+1) + 1]! - 1.