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\(A=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}.\text{ CMR : }\frac{7}{12}< A< \frac{5}{6}\)
Ta có :
\(A=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{99.100}\)
\(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}\)
\(A=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+...+\frac{1}{99}+\frac{1}{100}-2.\frac{1}{2}-2.\frac{1}{4}-...-2.\frac{1}{98}\)
\(A=1+...+\frac{1}{100}-1-\frac{1}{2}-\frac{1}{3}-...-\frac{1}{49}\)
\(A=\frac{1}{51}+...+\frac{1}{100}\)
\(\Rightarrow A< \frac{1}{51.25}=\frac{25}{51}< \frac{25}{30}=\frac{5}{6}\) (đpcm)
Và \(A>25.\frac{1}{75}+25.\frac{1}{100}=\frac{7}{12}\)
Ta có : A = 1 / (1.2) + 1 / (3.4) + ... + 1 / (99.100) > 1 / (1.2) + 1 / (3.4) = 1 / 2 + 1 / 12 = 7 / 12 (1)
Lại có : 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)
Từ (1) và (2) => 7 / 12 < A < 5 / 6
\(\forall n\in N;n\ne0\) Ta có : \(\frac{1}{n}-\frac{1}{n+1}-\frac{1}{n\left(n+1\right)}=\frac{\left(n+1\right)-n-1}{n\left(n+1\right)}=\frac{0}{\left(n+1\right)n}=0\)
\(\Rightarrow\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}=\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}+2\left[\frac{1}{n}-\frac{1}{n+1}-\frac{1}{n\left(n+1\right)}\right]}\)
\(=\sqrt{\left(1+\frac{1}{n}-\frac{1}{n+1}\right)^2}=1+\frac{1}{n}-\frac{1}{n+1}\)
Áp dụng ta được :
\(A=1+\frac{1}{2}-\frac{1}{3}+1+\frac{1}{3}-\frac{1}{4}+.....+1+\frac{1}{1100}-\frac{1}{1101}\)
\(=1099+\left(\frac{1}{2}+\frac{1}{3}+....+\frac{1}{1100}\right)-\left(\frac{1}{3}+\frac{1}{4}+....+\frac{1}{1101}\right)\)
\(=1099+\frac{1}{2}-\frac{1}{1101}=\frac{2421097}{2202}\)
\(\left(1-\frac{1}{6}\right)x\left(1-\frac{1}{7}\right)x\left(1-\frac{1}{8}\right)x\left(1-\frac{1}{9}\right)x\left(1-\frac{1}{10}\right)\)
\(=\frac{5}{6}x\frac{6}{7}x\frac{7}{8}x\frac{8}{9}x\frac{9}{10}\)
\(=\frac{1}{2}\)
=(1/1-1/6)x(1/1-1/7)x(1/1-1/8)x(1/1-19)x(1/1-1/10)
=5/6x6/6x7/8x8/9x9/10
\(=\frac{6}{5}\times\frac{7}{6}\times...\times\frac{11}{10}\)(lại lỗi đề)
\(=\frac{6×7×...×11}{5×6×...×10}\)
\(=\frac{11}{5}\)
\(1\frac{1}{5}\cdot1\frac{1}{6}\cdot1\frac{1}{7}\cdot1\frac{1}{8}\cdot1\frac{1}{9}\cdot1\frac{1}{10}\)
\(=\frac{6}{5}\cdot\frac{7}{6}\cdot\frac{8}{7}\cdot\frac{9}{8}\cdot\frac{10}{9}\cdot\frac{11}{10}\)
\(=\frac{6\cdot7\cdot8\cdot9\cdot10\cdot11}{5\cdot6\cdot7\cdot8\cdot9\cdot10}\)
\(=\frac{11}{5}\)