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\(\dfrac{1}{1\cdot6}+\dfrac{1}{6\cdot11}+\dfrac{1}{11\cdot16}+...+\dfrac{1}{\left(5n+1\right)\left(5n+6\right)}=\dfrac{n+1}{5n+6}\)
\(VT=\dfrac{1}{5}\left(\dfrac{5}{1\cdot6}+\dfrac{5}{6\cdot11}+...+\dfrac{5}{\left(5n+1\right)\left(5n+6\right)}\right)\)
\(=\dfrac{1}{5}\left(1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+...+\dfrac{1}{5n+1}-\dfrac{1}{5n+6}\right)\)
\(=\dfrac{1}{5}\left(1-\dfrac{1}{5n+6}\right)\)\(=\dfrac{1}{5}\cdot\left(\dfrac{5n+6}{5n+6}-\dfrac{1}{5n+6}\right)\)
\(=\dfrac{1}{5}\cdot\dfrac{5\left(n+1\right)}{5n+6}=\dfrac{n+1}{5n+6}=VP\)
Đặt :
\(A=\dfrac{3}{9.14}+\dfrac{3}{14.19}+......................+\dfrac{3}{\left(5n-1\right)\left(5n+4\right)}\)
\(A.\dfrac{5}{3}=\dfrac{5}{9.14}+\dfrac{5}{14.19}+..................+\dfrac{5}{\left(5n-1\right)\left(5n+1\right)}\)
\(A.\dfrac{5}{3}=\dfrac{1}{9}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{19}+..................+\dfrac{1}{5n-1}-\dfrac{1}{5n+4}\)
\(A.\dfrac{5}{3}=\dfrac{1}{9}-\dfrac{1}{5n+4}\)
\(A=\left(\dfrac{1}{9}-\dfrac{1}{5n+4}\right):\dfrac{3}{5}\)
\(A=\left(\dfrac{1}{9}-\dfrac{1}{5n+\text{4}}\right).\dfrac{3}{5}\)
\(A=\dfrac{1}{9}.\dfrac{3}{5}-\dfrac{1}{5n+4}.\dfrac{3}{5}\)
\(A=\dfrac{1}{15}-\dfrac{1}{5.\left(5n+4\right)}\)
\(\Rightarrow A< \dfrac{1}{15}\)
\(\Rightarrowđpcm\)
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\(VT=\dfrac{1}{5}\left(\dfrac{5}{1\cdot6}+\dfrac{5}{6\cdot11}+...+\dfrac{5}{\left(5n+1\right)\left(5n+6\right)}\right)\)
\(=\dfrac{1}{5}\left(1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+\dfrac{1}{11}-...+\dfrac{1}{5n+1}-\dfrac{1}{5n+6}\right)\)
\(=\dfrac{1}{5}\left(1-\dfrac{1}{5n+6}\right)\)
\(=\dfrac{1}{5}\cdot\dfrac{5n+6-1}{5n+6}\)
\(=\dfrac{n+1}{5n+6}=VP\)