Question

A=\(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+.................+\frac{1}{99.100}\)

CMR: 7/12< A< 5/6

 

Answer 1

A = 1/1.2 + 1/3.4 + 1/5.6 + ... + 1/99.100

A = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... + 1/99 - 1/100

A = ( 1 + 1/3 + 1/5 + ... + 1/99) - ( 1/2 + 1/4 + 1/6 + ... + 1/100)

A = ( 1 + 1/2 + 1/3 + 1/4 + .... + 1/99 + 1/100) - 2.( 1/2 + 1/4 + 1/6 + ....+ 1/100)

A = ( 1 + 1/2 + 1/3 + 1/4 + .... + 1/99 + 1/100) - ( 1 + 1/2 + 1/3 + ... + 1/50)

A = 1/51 + 1/52 + .... + 1/100

A = ( 1/51 + 1/52 + ... + 1/75) + ( 1/76 + 1/77 + ... + 1/100)

A > 1/75 . 25 + 1/100 . 25

A > 1/3 + 1/4 = 7/12 (1)

A < 1/50 . 25 + 1/75 x 25

a < 1/2 + 1/3 = 5/6 (2)

Từ (1) và (2) => 7/12 < A < 5/6

Chứng tỏ 7/12 < A < 5/6

Answer 2

A = 1/1.2 + 1/3.4 + 1/5.6 + ... + 1/99.100

A = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... + 1/99 - 1/100

A = ( 1 + 1/3 + 1/5 + ... + 1/99) - ( 1/2 + 1/4 + 1/6 + ... + 1/100)

A = ( 1 + 1/2 + 1/3 + 1/4 + .... + 1/99 + 1/100) - 2.( 1/2 + 1/4 + 1/6 + ....+ 1/100)

A = ( 1 + 1/2 + 1/3 + 1/4 + .... + 1/99 + 1/100) - ( 1 + 1/2 + 1/3 + ... + 1/50)

A = 1/51 + 1/52 + .... + 1/100

A = ( 1/51 + 1/52 + ... + 1/75) + ( 1/76 + 1/77 + ... + 1/100)

A > 1/75 . 25 + 1/100 . 25

A > 1/3 + 1/4 = 7/12 (1)

A < 1/50 . 25 + 1/75 x 25

a < 1/2 + 1/3 = 5/6 (2)

Từ (1) và (2) => 7/12 < A < 5/6

Chứng tỏ 7/12 < A < 5/6

Answer 3

A = 1/1.2 + 1/3.4 + 1/5.6 + ... + 1/99.100

A = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... + 1/99 - 1/100

A = ( 1 + 1/3 + 1/5 + ... + 1/99) - ( 1/2 + 1/4 + 1/6 + ... + 1/100)

A = ( 1 + 1/2 + 1/3 + 1/4 + .... + 1/99 + 1/100) - 2.( 1/2 + 1/4 + 1/6 + ....+ 1/100)

A = ( 1 + 1/2 + 1/3 + 1/4 + .... + 1/99 + 1/100) - ( 1 + 1/2 + 1/3 + ... + 1/50)

A = 1/51 + 1/52 + .... + 1/100

A = ( 1/51 + 1/52 + ... + 1/75) + ( 1/76 + 1/77 + ... + 1/100)

A > 1/75 . 25 + 1/100 . 25

A > 1/3 + 1/4 = 7/12 (1)

A < 1/50 . 25 + 1/75 x 25

a < 1/2 + 1/3 = 5/6 (2)

Từ (1) và (2) => 7/12 < A < 5/6

Chứng tỏ 7/12 < A < 5/6