\(=\frac{5}{3}.\left(\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{301.304}\right)\)

\(=\frac{5}{3}.\left(\frac{1}{4}-\frac{1}{304}\right)\)

\(=\frac{5}{3}.\frac{75}{304}\)

\(=\frac{125}{304}\)

\(\frac{5}{4×7}+\frac{5}{7×10}+\frac{5}{10×13}+...+\frac{5}{301×304}\)

\(=\frac{5}{4}-\frac{5}{7}+\frac{5}{7}-\frac{5}{10}+\frac{5}{10}-\frac{5}{13}+...+\frac{5}{301}-\frac{5}{304}\)

\(=\frac{5}{4}-\frac{5}{304}\)

\(=\frac{380}{304}-\frac{5}{304}\)

\(=\frac{375}{304}\)

Cbht

\(\frac{5}{4.7}+\frac{5}{7.10}+\frac{5}{10.13}+...+\frac{5}{301.304}\)

\(=\frac{5}{3}.\left(\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{301.304}\right)\)

\(=\frac{5}{3}.\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{301}-\frac{1}{304}\right)\)

\(=\frac{5}{3}.\left(\frac{1}{4}-\frac{1}{304}\right)\)

\(=\frac{5}{3}.\frac{75}{304}\)

\(=\frac{125}{304}\)

E= 7/4x7 + 7/7x10 =7/10x13+...+ 7/301x304

\(=\frac{7}{3}\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{301}-\frac{1}{304}\right)\)

\(=\frac{7}{3}\left(\frac{1}{4}-\frac{1}{304}\right)\)

\(=\frac{7}{3}\cdot\frac{75}{304}\)

\(=\frac{175}{304}\)

E = \(\frac{7}{4.7}+\frac{7}{7.10}+\frac{7}{10.13}+...+\frac{7}{301.304}\) 

   =\(\frac{7}{3}.\frac{7-4}{4.7}+\frac{7}{3}.\frac{10-7}{7.10}+\frac{7}{3}.\frac{13-10}{10.13}+...+\frac{7}{3}.\frac{304-301}{301.304}\)

 = \(\frac{7}{3}.\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+\frac{1}{10}-\frac{1}{13}+...+\frac{1}{301}-\frac{1}{304}\right)\)=\(\frac{7}{3}.\left(\frac{1}{4}-\frac{1}{304}\right)=\frac{7}{3}.\frac{75}{304}=\frac{175}{304}\)

\(\dfrac{5}{4\times7}+\dfrac{5}{7\times10}+....+\dfrac{5}{25\times28}+\dfrac{5}{28\times31}\)

\(=5\times\left(\dfrac{1}{4\times7}+\dfrac{1}{7\times10}+...+\dfrac{1}{28\times31}\right)\)

\(=\dfrac{5}{3}\times\left(\dfrac{3}{4\times7}+\dfrac{3}{7\times10}+....+\dfrac{3}{28\times31}\right)\)

\(=\dfrac{5}{3}\times\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{28}-\dfrac{1}{31}\right)\)

\(=\dfrac{5}{3}\times\left(\dfrac{1}{4}-\dfrac{1}{31}\right)\)

\(=\dfrac{5}{3}\times\dfrac{27}{124}\)

\(=\dfrac{45}{124}\)

`#3107`

\(\dfrac{5}{4\times7}+\dfrac{5}{7\times10}+...+\dfrac{5}{25\times28}+\dfrac{5}{28\times31}\)

\(=\dfrac{5}{3}\times\left(\dfrac{3}{4\times7}+\dfrac{3}{7\times10}+...+\dfrac{3}{25\times28}+\dfrac{3}{28\times31}\right)\)

\(=\dfrac{5}{3}\times\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{28}-\dfrac{1}{31}\right)\\ =\dfrac{5}{3}\times\left(\dfrac{1}{4}-\dfrac{1}{31}\right)\\ =\dfrac{5}{3}\times\dfrac{27}{124}\\ =\dfrac{45}{124}\)

\(C=\dfrac{1}{4.7}+\dfrac{1}{7.10}+\dfrac{1}{10.13}+...+\dfrac{1}{2020+2023}\)

\(=\dfrac{1}{3}\left(\dfrac{3}{4.7}+\dfrac{3}{7.10}+\dfrac{3}{10.13}+...+\dfrac{3}{2020.2023}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{13}+...+\dfrac{1}{2020}-\dfrac{1}{2023}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{4}-\dfrac{1}{2023}\right)\)

\(=\dfrac{1}{3}.\dfrac{2019}{8092}\)

\(=\dfrac{673}{8092}\)

=1/1-1/4+1/4-1/7+1/7-1/10+1/10-1/13+1/13-1/16

=1-1/16=15/16

1/1.4+1/4.7+1/7.10+1/10.13+1/13.16

=1/3.(3/1.4+3/4.7+3/7.10+3/10.13+3/13.16)

=1/3.(1/1-1/4+1/4-1/7+1/7-1/10+1/10-1/13+1/13-1/16)

=1/3.(1/1-1/16)

=1/3.(16/16-1/16)=1/3.15/16=5/16