求证一道数学归纳法的证明题
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求证一道数学归纳法的证明题
1·n+2(n-1)+...+(n-1)2+n·1=1/6·n(n+1)(n+2)
1·n+2(n-1)+...+(n-1)2+n·1=1/6·n(n+1)(n+2)
![求证一道数学归纳法的证明题](/uploads/image/z/15506630-62-0.jpg?t=%E6%B1%82%E8%AF%81%E4%B8%80%E9%81%93%E6%95%B0%E5%AD%A6%E5%BD%92%E7%BA%B3%E6%B3%95%E7%9A%84%E8%AF%81%E6%98%8E%E9%A2%98)
1·n+2(n-1)+...+(n-1)2+n·1=1/6·n(n+1)(n+2)
a1=n=1=1/6*1*2*3
a2=3n-2=4=1/6*2*3*4
a3=6n-8=10=1/6*3*4*5
……
an=1/6*n(n+1)(n+2)
an+1=1·(n+1)+2n+...+2n+(n+1)·1
=n+1+2(n-1)+2+3(n-2)+3+……+2(n-2)+2+n+1
=1/6*n(n+1)(n+2)+(1+2+3+……+2+1)
=1/6*n(n+1)(n+2)+1/2*(n+1)(n+2)
=1/6*(n+1)(n+2)(n+3)
原题得证
a1=n=1=1/6*1*2*3
a2=3n-2=4=1/6*2*3*4
a3=6n-8=10=1/6*3*4*5
……
an=1/6*n(n+1)(n+2)
an+1=1·(n+1)+2n+...+2n+(n+1)·1
=n+1+2(n-1)+2+3(n-2)+3+……+2(n-2)+2+n+1
=1/6*n(n+1)(n+2)+(1+2+3+……+2+1)
=1/6*n(n+1)(n+2)+1/2*(n+1)(n+2)
=1/6*(n+1)(n+2)(n+3)
原题得证