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\(A=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+....+\frac{10}{5^{10}}+\frac{11}{5^{11}}\)
\(\Rightarrow5A=1+\frac{2}{5}+\frac{3}{5^2}+....+\frac{10}{5^9}+\frac{11}{5^{10}}\)
\(\Rightarrow5A-A=\left(1+\frac{2}{5}+...+\frac{11}{5^{10}}\right)-\left(\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\right)\)
\(\Rightarrow4A=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{10}}-\frac{11}{5^{11}}\)(1)
Đặt \(B=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{10}}\)
\(\Rightarrow5B=5+1+\frac{1}{5}+...+\frac{1}{5^9}\)
\(\Rightarrow5B-B=\left(5+1+...+\frac{1}{5^9}\right)-\left(1+\frac{1}{5}+...+\frac{1}{5^{10}}\right)\)
\(\Rightarrow4B=5-\frac{1}{5^{10}}< 5\)
\(\Rightarrow B< \frac{5}{4}\)(2)
Thay (2) vào (1) \(\Rightarrow4A< \frac{5}{4}-\frac{11}{5^{11}}< \frac{5}{4}\)
\(\Rightarrow A< \frac{5}{16}\left(đpcm\right)\)
\(A=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+...+\frac{11}{5^{11}}\Rightarrow5A=1+\frac{2}{5}+\frac{3}{5^2}+...+\frac{11}{5^{10}}\)
\(\Rightarrow4A=5A-A=1+\left(\frac{1}{5}+\frac{1}{5^2}+\frac{...1}{5^{10}}\right)-\frac{11}{5^{11}}\)
\(< 1+\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{10}}\right)< 1+\frac{1}{4}=\frac{5}{4}\)
\(\Rightarrow A< \frac{5}{4}:4=\frac{5}{16}\)
Lưu ý : \(M=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{10}}\Rightarrow5M=1+\frac{1}{5}+...+\frac{1}{5^9}\Rightarrow4M=5M-M=1-\frac{1}{5^{10}}\)
\(\Rightarrow M=\frac{1}{4}-\frac{1}{5^{10}}:4< \frac{1}{4}\)
