Question

证明阶乘公式1×1!+2×2!+3×3!+...+(n-1)×(n-1)!=n!-1

Answer 1

用数学归纳法:

(1)当n=2时,左边=1=右边

(2)假设当n=k时,有1×1!+2×2!+3×3!+...+(k-1)×(k-1)!=k!-1

则当n=k+1时,

1×1!+2×2!+3×3!+...+(k-1)×(k-1)!+k*k!

=k!-1 +k*k!

=(k+1)k!-1

=(k+1)!-1

所以,当n=k+1时,命题成立

综上,原命题成立