若S=1*-2*+3*-4*+.+99*-100*,求S的值.PS：*指的是2次方.

Question

若S=1*-2*+3*-4*+.+99*-100*,求S的值.PS：*指的是2次方.

Answer 1

1²+2²+……+n²=Sn = n*(n + 1)*(2n + 1)/6
1+2+……+n=n(n+1)/2
S=1*-2*+3*-4*+.+99*-100*
=-[2x(2-1)+4x(4-1)+6x(6-1)+……+100x(100-1)]
=-[(2²+4²+……+100²)-(2+4+6+……+100)]
=-[4(1²+2²+……+50²)-2x(1+2+……+50)]
=-[4x50x51x101-2x50x(50+1)/2]
=--1027650