Tính tổng: 1 + 2² + 3² + ... + 99² + 100².

Đặt A = 1 + 2² + 3² + ... + 99² + 100².

= 1.1 + 2.2 + 3.3 + ... + 99.99 + 100.100.

= 1.(2 – 1) + 2.(3 – 1) + 3.(4 – 1) + ... + 99.(100 – 1) + 100.(101 – 1).

= 1.2 – 1 + 2.3 – 2 + 3.4 – 3 + ... + 99.100 – 1.99 + 100.101 – 100.

= (1.2 + 2.3 + 3.4 + ... + 99.100 + 100.101) – (1 + 2 + 3 + ... + 100).

Đặt B = 1.2 + 2.3 + 3.4 + ... + 99.100 + 100.101.

Suy ra 3B = 1.2.3 + 2.3.3 + 3.4.3 + ... + 99.100.3 + 100.101.3.

= 1.2.3 + 2.3.(4 – 1) + 3.4.(5 – 2) + ... + 99.100.(101 – 98) + 100.101.(102 – 99).

= 1.2.3 + 2.3.4 – 1.2.3 + 3.4.5 – 2.3.4 + ... + 99.100.101 – 98.99.100 + 100.101.102 – 99.100.101.

= 100.101.102.

Suy ra B = 100.101.102 : 3 = 100.101.34 = 343 400.

Đặt C = 1 + 2 + 3 + ... + 100 = (1 + 100) × 100 : 2 = 5050.

Vậy A = B – C = 343 400 – 5050 = 338 350.