a) Đề phải là thế này chứ \(\frac{5}{6.11}+\frac{5}{11.16}+\frac{5}{16.21}+...+\frac{5}{101.106}\)
Giai
\(=\frac{5}{6.11}+\frac{5}{11.16}+\frac{5}{16.21}+...+\frac{5}{101.106}\)
\(=\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+\frac{1}{16}-\frac{1}{21}+...+\frac{1}{101}-\frac{1}{106}\)
\(=\frac{1}{6}-\frac{1}{106}\)
\(=\frac{25}{159}\)
b) Đặt \(A=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{100}}\)
\(\Rightarrow5A=1+\frac{1}{5}+...+\frac{1}{5^{99}}\)
\(\Rightarrow5A-A=\left(1+\frac{1}{5}+...+\frac{1}{5^{99}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{100}}\right)\)
\(\Rightarrow4A=1-\frac{1}{5^{100}}\)
\(\Rightarrow A=\frac{1-\frac{1}{5^{100}}}{4}\)