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222-9+11+12:2*14+14 = ? ( )

## Toán Lớp 7: Chứng minh rằng `1/2 . 3/4 . 5/6 – 99/100 < 1/10`

Toán Lớp 7: Chứng minh rằng 1/2 . 3/4 . 5/6 …. 99/100 < 1/10

## Comments ( 2 )

1. Đặt A = 1/2 . 3/4 . 5/6 …. 99/100
Ta có : 1/2 < 2/3 , 3/4 < 4/5 , 5/6 < 6/7, …. , 99/100 < 100/101
⇒ 1/2 . 3/4 . 5/6 …. 99/100 < 2/3 . 4/5 . 6/7 …. 100/101
⇒ S < 2/3 . 4/5 . 6/7 …. 100/101
⇒ S^2 < ( 1/2 . 3/4 . 5/6 …. 99/100 ) . ( 2/3 . 4/5 . 6/7 …. 100/101 )
⇒ S^2 < 1/2 . 2/3 . 3/4 . 4/5 …. 99/100 . 100/101
⇒ S^2 < ( 1 . 2 . 3 . 4 …. 99 . 100 )/( 2 . 3 . 4 . 5 …. 101 )
⇒ S^2 < 1/101 < 1/100
Do S^2 < 1/100 ⇒ S < 1/10
⇒ 1/2 . 3/4 . 5/6 …. 99/100 < 1/10 ( Điều phải chứng minh )

2. Đặt:
A = 1/2 . 3/4 . 5/6 . …. . 99/100
Có:
1/2 < 2/3
3/4 < 4/5
5/6 < 6/7
99/100 < 100/101
=> A < 2/3 . 4/5 . 6/7 . … . 100/101
=> A . A < (1/2 . 3/4 . 5/6 . …. . 99/100) . (2/3 . 4/5 . 6/7 . … . 100/101)
=> A^2 < 1/2 . 2/3 . 3/4 . 4/5 . … . 100/101
=> A^2 < 1/100
=> A < 1/10 (Đpcm)
Vậy 1/2 . 3/4 . 5/6 . …. . 99/100 < 1/10