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Câu 1:
\(S=\frac{10}{7}+\frac{10}{7^2}+\frac{10}{7^3}+...+\frac{10}{7^{10}}\)
\(\frac{1}{7}S=\frac{10}{7^2}+\frac{10}{7^3}+....+\frac{10}{7^{11}}\)
\(\rightarrow\)\(\left(1-\frac{1}{7}\right).S=\frac{10}{7}-\frac{10}{7^{11}}\)
=> \(S=\frac{10.7^{10}-10}{7^{10}.6}\)
a) \(A=\frac{-7}{813}+496.\left(\frac{-7}{813}\right)+\left(\frac{-7}{813}\right).316\)
\(=\frac{-7}{813}.\left(1+496+316\right)\)
\(=\frac{-7}{813}.813\)
\(=-7\)
b) \(B=\frac{-9}{10}.\frac{5}{14}+\frac{1}{10}.\left(\frac{-9}{2}\right)+\frac{1}{7}.\left(\frac{-9}{10}\right)\)
\(=\frac{-9}{10}.\left(\frac{5}{14}+\frac{1}{2}+\frac{1}{7}\right)\)
\(=\frac{-9}{10}.1\)
\(=\frac{-9}{10}\)
1/10 A =7/10^2+7/10^3+..............+7/10^2020
9/10*A=(7/10+7/10^2+......................+7/10^2019)-(7/10^2+7/10^3+........+7/10^2020)
=7/10-7/10^2020
A=10/9 .(7/10-7/10^2020)
\(M=\frac{32}{323}\) \(N=\frac{86}{589}\) \(\frac{M}{N}=\frac{496}{731}\)
\(\frac{-8}{10}.\frac{74}{7}+\frac{17}{7}.\frac{-8}{13}+\frac{3}{5}=\frac{-8}{7}.\frac{37}{5}+\frac{-8}{7}.\frac{17}{13}+\frac{3}{5}\)
\(=\frac{-8}{7}.\left(\frac{37}{5}+\frac{17}{13}\right)+\frac{3}{5}=\frac{-8}{7}.\frac{566}{65}+\frac{3}{5}\)
\(=\frac{-4528}{455}+\frac{273}{455}=\frac{-4255}{455}=\frac{851}{91}\)
\(A=\frac{7}{10}+\frac{7}{10^2}+...+\frac{7}{10^{100}}\)
\(10A=7+\frac{7}{10}+...+\frac{7}{10^{99}}\)
\(\Rightarrow10A-A=9A=7-\frac{7}{10^{100}}\)
Ta có : \(10A=7+\frac{7}{10}+\frac{7}{10^2}+...+\frac{7}{10^{99}}\)
\(A=\frac{7}{10}+\frac{7}{10^2}+...+\frac{7}{10^{99}}+\frac{7}{10^{100}}\)
\(\Rightarrow9A=10A-A=7-\frac{7}{10^{100}}\)
\(\Rightarrow A=\frac{7-\frac{7}{10^{100}}}{9}\)