Question

Cho tổng A = 1 10 + 1 11 + 1 12 + ... + 1 99 + 1 100 . Chứng tỏ rằng A > 1

Answer 1

A = 1 10 + 1 11 + 1 12 + ... + 1 99 + 1 100 = 1 10 + 1 11 + 1 12 + ... + 1 90 + 1 91 + 1 92 + ... 1 99 + 1 100

Đặt A 1 = 1 10 + 1 11 + 1 12 + ... + 1 90 ;  A 2 = 1 91 + 1 92 + ... 1 99 + 1 100

Ta có  1 10 > 1 90 ;   1 11 > 1 90 ;   1 12 > 1 90 ; ... ; 1 89 > 1 90

A 1 = 1 10 + 1 11 + 1 12 + ... + 1 90 > 81 ⋅ 1 90

A 1 = 1 10 + 1 11 + 1 12 + ... + 1 90 > 81 ⋅ 1 90

Tương tự  1 91 > 1 100 ;   1 92 > 1 100 ;   ... ; 1 99 > 1 100

A 2 = 1 91 + 1 92 + ... 1 99 + 1 100 > 10 ⋅ 1 100

A 2 = 1 91 + 1 92 + ... 1 99 + 1 100 > 10 100 = 1 10

A = 1 10 + 1 11 + 1 12 + ... + 1 99 + 1 100 = A 1 + A 2 > 9 10 + 1 10

A = 1 10 + 1 11 + 1 12 + ... + 1 99 + 1 100 > 1