Tính B = 1.2² + 2.3² + … + 99.100².

B = 1.2² + 2.3² + … + 99.100²

B = 1.2.2 + 2.3.3 + … + 99.100.100

B = 1.2.(3–1) + 2.3.(4–1) + …. + 99.100.(101 – 1)

B = 1.2.3 – 1.2 + 2.3.4 – 1.2.3 + … + 99.100.101 – 99.100

B = (1.2.3 + 2.3.4 + … + 99.100.101) – (1.2 + 2.3 + … + 99.100)

B = A – C

+ Xét A = 1.2.3 + 2.3.4 + … + 99.100.101

4A = 1.2.3.4 + 2.3.4.(5–1) + ... + 99.100.101.(102–98)

4A = 1.2.3.4 + 2.3.4.5 – 1.2.3.4 + ... + 99.100.101.102 – 98.99.100.101

4A = 99.100.101.102

A = 25497450

+ Xét C = 1.2 + 2.3 + … + 99.100

3C = 1.2.3 + 2.3.3 + ….+ 99.100.3

3C = 1.2.3 + 2.3.(4–1) + … +99.100.(101–98)

3C = 99.100.101

C = 333300

Suy ra: B = A – C = 25497450 – 333300 = 25164150.