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Ta có :
A = \(\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
A = \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
A = \(\left(1+\frac{1}{3}+...+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
A = \(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}+\frac{1}{100}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
A = \(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}+\frac{1}{100}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)\)
A = \(\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+...+\frac{1}{100}\)
Tách A thành 2 nhóm,ta được :
A = \(\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}\right)+\left(\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}\right)\)
Lại có : \(\frac{1}{51}>\frac{1}{52}>...>\frac{1}{75}\text{ }\text{ }\)
\(\frac{1}{76}>\frac{1}{77}>...>\frac{1}{100}\text{ }\text{ }\)
A > \(\left(\frac{1}{75}+\frac{1}{75}+...+\frac{1}{75}\right)+\left(\frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}\right)=\frac{1}{75}.25+\frac{1}{100}.25\)
\(=\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)
A < \(\left(\frac{1}{51}+\frac{1}{51}+...+\frac{1}{51}\right)+\left(\frac{1}{76}+\frac{1}{76}+...+\frac{1}{76}\right)=\frac{1}{51}.25+\frac{1}{76}.25< \frac{1}{50}.25+\frac{1}{75}.25\)
\(=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)
Vậy \(\frac{7}{12}< A< \frac{5}{6}\)
Ta thấy : A = 1 - 1/2 + 1/3 - 1/4 +...+ 1/99 +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 + 1/53 +...+ 1/100
Do đó : A = (1/51 + 1/52 + 1/53 +...+ 1/75) + (1/76 + 1/77 +...+ 1/100)
Ta có : 1/51 > 1/52 > ... > 1/75 ; 1/76 > 1/77 > ... > 1/100 nên :
A > 1/75.25 + 1/100.25 = 1/3 + 1/4 = 7/12
A < 1/51.25 < 1/50.25 + 1/75.25 = 1/2 +1/3 = 5/6
Vậy 7/12 < A <5/6 . ^_^
\(A=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}=\left(\frac{1}{1}+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
\(A=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{99}\right)+\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
\(A=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)=\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)
+) \(A=\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}\right)+\left(\frac{1}{76}+...+\frac{1}{100}\right)>\left(\frac{1}{75}+...+\frac{1}{75}\right)+\left(\frac{1}{100}+...+\frac{1}{100}\right)\)
=> \(A>\frac{25}{75}+\frac{25}{100}=\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)
+) \(A=\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}\right)+\left(\frac{1}{76}+...+\frac{1}{100}\right)<\left(\frac{1}{50}+...+\frac{1}{50}\right)+\left(\frac{1}{75}+...+\frac{1}{75}\right)\)
=> \(A<\frac{25}{50}+\frac{25}{100}=\frac{1}{2}+\frac{1}{4}=\frac{3}{4}<\frac{5}{6}\)
Vậy...
Chứng minh : \(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{49.50}=\frac{1}{26}+\frac{1}{27}+\frac{1}{28}+...+\frac{1}{50}\): như câu trên
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 \(\frac{7}{12}< A< \frac{5}{6}\)
https://olm.vn/hoi-dap/question/119017.html
tham khảo ở đó nhé!!!
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
= 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
Câu hỏi của Vũ Thị Kim Oanh - Toán lớp 7 - Học toán với OnlineMath
Tham khảo
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