1x2+2x3+3x4+4x5........99x100

Question

谁会简便方法的帮我一下。谢谢了！

Answer 1

1x2+2x3+3x4+4x5........99x100

=1/3*1*2*3+1/3*[2*3*4-1*2*3]+1/3[3*4*5-2*3*4]+...+1/3[99*100*101-98*99*100]

=1/3[1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+....+99*100*101-98*99*100]

=1/3*99*100*101

=3300*101

=333300

公式：Sn=1*2+2*3+3*4+...+n(n+1)=n(n+1)(n+2)/3

Answer 2

最简洁明了的应是：
∑(k-1)k=∑k^2 - ∑k=(n-1)n(n+1)/3,(k=1,2,3...n)

把100代入上式

1x2+2x3+3x4+4x5+5x6+...+99x100=99*100*101/3= 333300

Answer 3

设（1X2)/2+(2X3)/2+(3X4)/2+.………………+(n+1)n/2=S
得1x2+2x3+3x4+………………+(n+1)n=2S
由n(n-1)+(n+1)n=n^2-n+n^2+n=2n^2
当n为偶数时S=2^2+4^2+6^2+8^2+………………+n^2
当n为奇数时S=2^2+4^2+6^2+8^2+………………+(n-1)^2+(n+1)n/2
当n为偶数时
假设M=2^2+4^2+6^2+8^2+………………+n^2
=4(1^2+2^2+3^2+4^2+………………+(n/2)^2)
=4*(1/6)(n/2)(n/2+1)(n+1)
=(1/6)n(n+1)(n+2)
所以我们可以得出结论
当n为偶数时
(1X2)/2+(2X3)/2+(3X4)/2+.....................+(n+1)n/2=(1/6)n(n+1)(n+2)
当n为奇数时
(1X2)/2+(2X3)/2+(3X4)/2+.....................+(n+1)n/2=(1/6)(n-1)n(n+1)+(n+1)n/2

Answer 4

100(1*100)/2+2(1-3/2)/1-3/2