Ta có :

\(\frac{50}{111}>\frac{50}{200}\)

\(\frac{50}{112}>\frac{50}{200}\)

\(\frac{50}{113}>\frac{50}{200}\)

\(\frac{50}{114}>\frac{50}{200}\)

\(\Rightarrow A>\frac{50}{200}+\frac{50}{200}+\frac{50}{200}+\frac{50}{200}\)hay \(A>\frac{50}{200}.4\left(1\right)\)

Mặt khác :

\(\frac{50}{111}< \frac{50}{100}\)

\(\frac{50}{112}< \frac{50}{100}\)

\(\frac{50}{113}< \frac{50}{100}\)

\(\frac{50}{114}< \frac{50}{100}\)

\(\Rightarrow A< \frac{50}{100}+\frac{50}{100}+\frac{50}{100}+\frac{50}{100}\)hay \(A< \frac{50}{100}.4\left(2\right)\)

Từ \(\left(1\right)\)và \(\left(2\right)\Rightarrow1< A< 2\left(đpcm\right)\)

em lp 5 nen ko biet!

\(\frac{50}{111}>\frac{1}{4};\frac{50}{112}>\frac{1}{4};\frac{50}{113}>\frac{1}{4};\frac{50}{114}>\frac{1}{4}\)

\(A=\frac{50}{111}+\frac{50}{112}+\frac{50}{113}+\frac{50}{114}>\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1\)(1)

\(\frac{50}{111}< \frac{1}{2};\frac{50}{112}< \frac{1}{2};\frac{50}{113}< \frac{1}{2};\frac{50}{114}< \frac{1}{2}\)

\(\Rightarrow A=\frac{50}{111}+\frac{50}{112}+\frac{50}{113}+\frac{50}{114}< \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2\)(2)

từ (1) và (2) \(\Rightarrow1< A< 2\)

50/111 < 50/100

50/112 < 50/100

50/113 < 50/100 

50/114 < 50/100

=> A < 200/100 => A < 2

50/111 > 50/200

50/112 > 50/200

50/113 > 50/200

50/114 > 50/200

=> A > 200/200 => A > 1

Vậy 1 < A < 2

AI THẤY OK ỦNG HỘ NHÉ 

Ta có \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{7\cdot8}\)

\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-....+\frac{1}{7}-\frac{1}{8}\)

\(\Rightarrow A< 1-\frac{1}{8}< 1\)

A = 1/1² + 1/2² + 1/3² + ... + 1/50²

A = 1/1.1 + 1/2.2 + 1/3.3 + ... + 1/50.50

A < 1/1 + 1/1.2 + 1/2.3 + ... + 1/49.50

A < 1 + 1 - 1/2 + 1/2 - 1/3 + ... + 1/49 - 1/50

A < 2 - 1/50 < 2

Chứng tỏ A < 2

Đặt B=1/1+1/1.2+...+1/49.50

Ta có:

A=1/1^2+1/2^2+...+1/50^2<B=1/1+1/1.2+...+1/49.50 (1)

Mà B=1/1+1/1.2+...+1/49.50

=1+1-1/2+1/2-1/3+...+1/49-1/50

=2-1/50 <2 (2)

Từ (1) và (2) =>A<B<2

=>A<2

\(A< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{19\cdot20}\)

\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{20}\)

\(\Rightarrow A< 1-\frac{1}{20}< 1\)

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