Tính nhanh: 1.4 + 2.5 + 3.6 + ... + 100.103.

Đặt A = 1.4 + 2.5 + 3.6 + ... + 100.103.

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

= 1.2 + 2.3 + 3.4 + ... + 100.101 + (1.2 + 2.2 + 3.2 + ... + 100.2).

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

Đặt M = 1.2 + 2.3 + 3.4 + ... + 100.101.

Suy ra 3M = 1.2.3 + 2.3.3 + 3.4.3 + ... + 100.101.3.

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

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

= 100.101.102.

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

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

Vậy A = M + 2N = 343 400 + 2.5050 = 353 500.

Vậy A = 353 500.