Question

数学归纳法证明题

证明1×n＋2×(n－1)＋3(n－2)＋……n×1=1/6n(n+1)(n+2)，归纳法，谢谢大神

Answer 1

当n=1时
1×1=1/6×1(1+1)(1+2)成立
假设当n=k时
1×k+2×（k-1)+...k*1=1/6k(k+1)(k+2)成立
当n=k+1时

1×(k+1)+2×(k+1-1)+...+(k+1)×1= 1×k+2×（k-1)+....k*1+1+2+....+(k+1)

Answer 2

n=k+1时
1×(k+1)＋2×k＋3(k－1)＋……(k+1)×1
=1×k＋2×(k－1)＋3(k－2)＋……k×1+(1+2+3+...k+k+1)
=k(k+1)(k+2)/6+(1+k)(2+k)/2
=(k+1)(k+2)(k+3)/6