Gọi biểu thức trên là A, ta có :
A = 1x2 + 2x3 + 3x4 + 4x5 + ...+ 99x100
A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 99x100x3
A x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98)
A x 3 = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100.
A x 3 = 99x100x101
A = 99x100x101 : 3
A = 333300
Cho A = 1 x 2 + 2 x 3 + 3 x 4 +.................+ 98 x 99
Ta có: 3A = 3.1.2 + 3.2.3 + ... + 3.98.99
3A = (3 - 0).1.2 + (4 - 1).2.3 + ... + (100 - 97).98.99
3A = 3.1.2 - 0.1.2 + 4.2.3 - 1.2.3 + ... + 100.99.98 - 97.98.99
3A = -0.1.2 + 98.99.100
3A = 98.99.100
Vậy: A = 98.99.100 : 3 = 970200 : 3 = 323400
