Question

A=1/2+1/2^2+1/2^3+...+1/2^20

Chứng minh A<1

Chứng minh  3/1^2 x 2^2 + 5/2^3 x 3^2 + 7/3^2 x 4^2 + ... + 19/9^2 x 10^2 < 1

 

Answer 1

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Answer 2

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Answer 3

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Answer 4

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Answer 5

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Answer 6

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Answer 7

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Answer 8

`Answer:`

a. \(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}< 1\)

\(\Rightarrow2A=2.\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}\right)\)

\(\Rightarrow2A-A=\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{19}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}\right)\)

\(\Rightarrow A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{19}}-\frac{1}{2}-\frac{1}{2^2}-\frac{1}{2^3}-...-\frac{1}{2^{20}}\)

\(\Rightarrow A=1+\frac{1}{2}-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^2}+...+\frac{1}{2^{19}}-\frac{1}{2^{20}}\)

\(\Rightarrow A=1-\frac{1}{2^{20}}\)

\(\Rightarrow A< 1\)

b. \(\frac{3}{1^2.2^2}+\frac{5}{2^2.3^2}+\frac{7}{3^2.4^2}+...+\frac{19}{9^2.10^2}\)

\(=\left(\frac{1}{1^2}-\frac{1}{2^2}\right)+\left(\frac{1}{2^2}-\frac{1}{3^2}\right)+\left(\frac{1}{3^2}-\frac{1}{4^2}\right)+...+\left(\frac{1}{9^2}-\frac{1}{10^2}\right)\)

\(=\frac{1}{1}-\frac{1}{10^2}\)

\(=1-\frac{1}{100}< 1\)