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Trước hết ta biến đổi A thành \(\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+...+\frac{1}{100}\)

Do đó : \(A=\left[\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}\right]+\left[\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}\right]\)

Ta có : \(\frac{1}{51}>\frac{1}{52}>...>\frac{1}{75},\frac{1}{76}>\frac{1}{77}>...>\frac{1}{100}\)nên

\(A>\frac{1}{75}\cdot25+\frac{1}{100}\cdot25=\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)

\(A< \frac{1}{51}\cdot25+\frac{1}{76}\cdot25< \frac{1}{50}\cdot25+\frac{1}{75}\cdot25=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)

Vậy \(\frac{7}{12}< A< \frac{5}{6}\)

Cách biến đổi :

Đặt \(A=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)

Dễ thấy \(A=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}\)

\(A=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}+\frac{1}{100}-2\left[\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{100}\right]\)

\(=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}+\frac{1}{100}-\left[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right]\)

\(=\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)

\(\Rightarrow A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+............+\frac{1}{99}-\frac{1}{100}\)

\(\Rightarrow A=1-\frac{1}{100}\)

\(\Rightarrow A=\frac{99}{100}\)

Vì \(\frac{7}{12}< \frac{99}{100}< \frac{5}{6}\Rightarrow\frac{7}{12}< A< \frac{5}{6}\)               ĐPCM

( Bài này ko ai lm thì t lm cho )

Câu hỏi của Vũ Thị Kim Oanh - Toán lớp 7 - Học toán với OnlineMath

A = 1 / (1.2) + 1 / (3.4) + ... + 1 / (99.100) > 1 / (1.2) + 1 / (3.4) = 1 / 2 + 1 / 12 = 7 / 12 (1)
A = 1 / (1.2) + 1 / (3.4) + ... + 1 / (99.100) = (1 - 1 / 2) + (1 / 3 - 1 / 4) + ... + (1 / 99 - 100) = (1 - 1 / 2 + 1 / 3) - (1 / 4 - 1 / 5) - (1 / 6 - 1 / 7) - ... - (1 / 98 - 1 / 99) - 1 / 100 < 1 - 1 / 2 + 1 / 3 = 5 / 6                (2) 
(1), (2)  => 7 / 12 < A < 5 / 6

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

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

=(1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 +.....+ 1/99 + 1/100) - ( 1 + 1/2 + 1/3 + .... + 1/50)

= 1/51 + 1/52 + 1/53 +....+ 1/100....>1/100

= ( 1/51 + 1/52 + 1/53 +.....+ 1/75) + ( 1/76 + 1/77 + 1/78 +.....+ 1/100)

Có 1/51>1/52>1/53>....>1/75 ; 1/76>1/77>1/78>....>1/100

A> 1/75.25 + 1/100.25= 1/3 + 1/4 = 7/12

A< 1/51.25+ 1/76.25 < 1/50.25 + 1/75.25= 1/2+1/3=5/6

Vậy 7/12< A< 5/6

\(\frac{1}{1.2}+\frac{1}{3.4}+......+\frac{1}{49.50}=1-\frac{1}{2}+\frac{1}{3}-....+\frac{1}{49}-\frac{1}{50}=\left(1+\frac{1}{3}+....+\frac{1}{49}\right)-\left(\frac{1}{2}+\frac{1}{4}+.....+\frac{1}{50}\right)=\left(1+\frac{1}{2}+.....+\frac{1}{50}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{50}\right)=\left(1+\frac{1}{2}+....+\frac{1}{50}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{25}\right)=\frac{1}{26}+\frac{1}{27}+....+\frac{1}{50}\left(đpcm\right)\)

\(theocaua\Rightarrow A=\frac{1}{26}+\frac{1}{27}+......+\frac{1}{50}>\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\left(5sohang\right)+\frac{1}{40}+\frac{1}{40}+....+\frac{1}{40}\left(10sohang\right)+\frac{1}{50}+\frac{1}{50}+....+\frac{1}{50}\left(10sohang\right)=\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{37}{60}>\frac{35}{60}=\frac{7}{12}\left(1\right)\)

\(A=\frac{1}{26}+\frac{1}{27}+....+\frac{1}{50}< \frac{1}{25}+\frac{1}{25}+...+\frac{1}{25}\left(5sohang\right)+\frac{1}{30}+\frac{1}{30}+....+\frac{1}{30}\left(10sohang\right)+\frac{1}{40}+\frac{1}{40}+.....+\frac{1}{40}\left(10sohang\right)=\frac{1}{4}+\frac{1}{3}+\frac{1}{5}=\frac{47}{60}< \frac{5}{6}=\frac{50}{60}\left(2\right)\) \(\left(1\right);\left(2\right)\Rightarrow\frac{7}{12}< A< \frac{5}{6}\)

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

\(A=\frac{1}{2}-\frac{1}{12}+...+\frac{1}{99000}>\frac{1}{2}+\frac{1}{12}=\frac{7}{12}\)

\(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}\)

\(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-...-\frac{1}{98}-\frac{1}{99}-\frac{1}{100}< 1-\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)

\(\RightarrowĐPCM\)

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