\(=\dfrac{1}{7}+\dfrac{4}{6}\left(\dfrac{6}{7\cdot13}+\dfrac{6}{13\cdot19}+...+\dfrac{6}{43\cdot49}\right)\)

\(=\dfrac{1}{7}+\dfrac{2}{3}\left(\dfrac{1}{7}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{19}+...+\dfrac{1}{43}-\dfrac{1}{49}\right)\)

\(=\dfrac{1}{7}+\dfrac{2}{3}\cdot\dfrac{6}{49}=\dfrac{1}{7}+\dfrac{4}{49}=\dfrac{11}{49}\)

E = \(\frac{36}{1\cdot7}+\frac{36}{7\cdot13}+...+\frac{36}{94\cdot100}=\frac{36}{6}\left[\frac{1}{1\cdot7}+\frac{1}{7\cdot13}+...+\frac{1}{94\cdot100}\right]\)

\(=6\left[1-\frac{1}{7}+\frac{1}{7}-\frac{1}{13}+...+\frac{1}{94}-\frac{1}{100}\right]=6\left[1-\frac{1}{100}\right]\)

\(=6\cdot\frac{99}{100}=\frac{297}{50}\)

F = \(\frac{1}{10}+\frac{1}{40}+\frac{1}{88}+...+\frac{1}{\left[3a+2\right]\left[3a+5\right]}\)

\(=\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot11}+...+\frac{1}{\left[3a+2\right]\left[3a+5\right]}\)

\(=\frac{1}{3}\left[\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+...+\frac{1}{3a+2}-\frac{1}{3a+5}\right]\)

\(=\frac{1}{3}\left[\frac{1}{2}-\frac{1}{3a+5}\right]=\frac{1}{6}-\frac{1}{9a+15}\)

G = \(\frac{1}{2\cdot3}+\frac{2}{3\cdot5}+\frac{3}{5\cdot8}+\frac{4}{8\cdot12}+\frac{5}{12\cdot17}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{12}-\frac{1}{17}\)

\(=\frac{1}{2}-\frac{1}{17}=\frac{15}{34}\)

E=36/1-36/7+36/7-36/13+...+36/94-36/100

  =36-36/100=891/25

\(\frac{5}{1.7}+\frac{5}{7.13}+\frac{5}{13.19}+...+\frac{5}{2017.2023}\)

\(=5.\frac{1}{6}.\left(\frac{6}{1.7}+\frac{6}{7.13}+\frac{6}{13.19}+...+\frac{6}{2017.2023}\right)\)

\(=\frac{5}{6}.\left(1-\frac{1}{7}+\frac{1}{7}-\frac{1}{13}+\frac{1}{13}-\frac{1}{19}+...+\frac{1}{2017}-\frac{1}{2023}\right)\)

\(=\frac{5}{6}.\left(1-\frac{1}{2023}\right)\)

\(=\frac{5}{6}.\frac{2022}{2023}\)

\(=\frac{1685}{2023}\)

\(\frac{5}{1.7}+\frac{5}{7.13}+\frac{5}{13.19}+...+\frac{5}{2017.2023}\)

\(=\frac{5.6}{1.7.6}+\frac{5.6}{7.13.6}+\frac{5.6}{13.19.6}+.....+\frac{5.6}{2017.2023.6}\)

\(=\frac{5}{6}.\left(\frac{6}{1.7}+\frac{6}{7.13}+\frac{6}{13.19}+...+\frac{6}{2017.2023}\right)\)

\(=\frac{5}{6}.\left(1-\frac{1}{7}+\frac{1}{7}-\frac{1}{13}+\frac{1}{13}-\frac{1}{19}+...+\frac{1}{2017}-\frac{1}{2023}\right)\)

\(=\frac{5}{6}.\left(1-\frac{1}{2023}\right)\)

\(=\frac{5}{6}.\frac{2022}{2023}\)

\(=\frac{1685}{2023}\)

1/7+1/91+1/247+1/475+1/775+1/1147=? (1)
ta có: (1) <=>: 1/(1.7)+1/(7.13)+1/(13.19)+1/(19.25)+1/(25.31)+1/(31.37)
=1/6.(1-1/7+1/7-1/13+1/13-1/19+1/19-1/25+1/25-1/31+1/31-1/37)
=1/6.(1-1/37)=6/37

\(\frac{1}{1.7}+\frac{1}{7.13}+\frac{1}{13.19}+...+\frac{1}{61.67}\)

=6.\(\left(\frac{1}{1.7}+\frac{1}{7.13}+...+\frac{1}{61.67}\right)\):6

=\((\frac{6}{1.7}+\frac{6}{7.13}+...+\frac{6}{61.67}):6\)

=\(\left(1-\frac{1}{7}+\frac{1}{7}+\frac{1}{13}+...+\frac{1}{61}+\frac{1}{67}\right):6\)

=\(\left(1-\frac{1}{67}\right):6\)

=\(\frac{66}{67}:6=\frac{66}{67}.\frac{1}{6}=\frac{11}{67}\)