1x2+3x4+5x6+.+29X30=?2x3+4x5+...+20x21=?1x2+2x3+3x4+...20x21=?30²-29²+28²-27²+.+2²-1²=?

Question

1x2+3x4+5x6+.+29X30=?2x3+4x5+...+20x21=?1x2+2x3+3x4+...20x21=?
30²-29²+28²-27²+.+2²-1²=?

Answer 1

运用公式：1²+2²+3²+...+n²=n(n+1)(2n+1)/6
1x2+3x4+5x6+...+29x30
=2²-2 +4²-4 +6²-6 +...+30²-30
=2²+4²+6²+...+30²-(2+4+6+...+30)
=(2x1)²+(2x2)²+(2x3)²+...+(2x15)²-(2+30)x15/2
=4x(1²+2²+3²+...+15²)-240
=4x(15x16x31/6)-240
=4x1240-240
=4720
2x3+4x5+6x7+...+20x21
=2²+2 +4²+4 +6²+6 +...+20²+20
=2²+4²+6²+...+20²+(2+4+6+...+20)
=(2x1)²+(2x2)²+(2x3)²+...+(2x10)²+(2+20)x10/2
=4x(1²+2²+3²+...+10²)-110
=4x(10x11x21/6)-110
=4x385-110
=1430
1x2+2x3+3x4+...+20x21
=1²+1 +2²+2 +3²+3 +...+20²+20
=1²+2²+3²+...+20²+(1+2+3+...+20)
=(20x21x41/6)-(1+20)x20/2
=2870-210
=2660