Tính: A = 1.99^2 + 2.98^2 + 3.97^2 + … + 49.51^2

[Chuyên đề dãy số viết theo quy luật] – Đề bài: Tính: A = 1.99² + 2.98² + 3.97² + … + 49.51²

Giải:

Ta có: A = 1.99² + 2.98² + 3.97² + … + 49.51²

A = (100 – 99).99² + (100 – 98).98² + (100 – 97).97² + … + (100 – 51).51²

A = 100.(99² + 98² + 97² + … + 51²) – (99³ + 98³ + 97³ + … + 51³)

A = 100.(51² + 52² + 53² + … + 99²) – (51³ + 52³ + 53³ + … + 99³)

Đặt B = 51² + 52² + 53² + … + 99²

B = 50.51 + 51 + 51.52 + 52 + 52.53 + 53 + … + 98.99 + 99

B = (50.51 + 51.52 + 52.53 + … + 98.99) + (51 + 52 + 53 + … + 99)

B = (98.99.100 – 49.50.51) : 3 + (51 + 99).49 : 2

B = 49.50.(2.99.2 – 51) : 3 + 49.75

B = 49.50.115 + 49.75

B = 285 425

Đặt C = 51³ + 52³ + 53³ + … + 99³

C = 50.51.52 + 51 + 51.52.53 + 52 + 52.53.54 + 53 + … + 98.99.100 + 99

C = (50.51.52 + 51.52.53 + 52.53.54 + … + 98.99.100) + (51 + 52 + 53 + … + 99)

C = (98.99.100.101 – 49.50.51.52) : 4 + 49.75

C = 98.99.25.101 – 49.25.51.26 + 49.75

C = 22 776 875

=> A = 100B – C = 100.285425 – 22 776 875 = 5765625.