\(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)

\(=3.\left(\frac{1}{1.4}-\frac{1}{4.7}\right)+3.\left(\frac{1}{4.7}-\frac{1}{7.10}\right)+...+3.\left(\frac{1}{54.57}-\frac{1}{57.60}\right)\)

\(=3\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...-\frac{1}{54.57}+\frac{1}{54.57}-\frac{1}{57.60}\right)\)

\(=3\left(\frac{1}{1.4}-\frac{1}{57.60}\right)\)

\(=3\left(\frac{1}{4}-\frac{1}{3420}\right)\)

\(=3\left(\frac{855}{3420}-\frac{1}{3420}\right)\)

\(=3.\frac{427}{1710}\)

\(=\frac{427}{570}\)

=2.(6/1.4.7 + 6/4.7.10 + 6/7.10.13 + ... + 6/54.57.60)

=2.(1/1.4-1/4.7+1/4.7-1/7.10+1/7.10-1/10.13+...+1/54.57-1/57.60)

=2(1.4-1/57.60)

TỰ TÍNH

Ta có \(A=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+\frac{1}{7.10}-...+\frac{1}{16.19}-\frac{1}{19.22}\)

\(=\frac{1}{4}-\frac{1}{418}=\frac{207}{836}\)

\(A=\frac{6}{1\cdot4\cdot7}+\frac{6}{4\cdot7\cdot10}+\frac{6}{7\cdot10\cdot13}+...+\frac{6}{16\cdot19\cdot22}\)

\(A=\frac{1}{1\cdot4}-\frac{1}{4\cdot7}+\frac{1}{4\cdot7}-\frac{1}{7\cdot10}+...+\frac{1}{16\cdot19}-\frac{1}{19\cdot22}\)

\(A=\frac{1}{4}-\frac{1}{19\cdot22}=\frac{207}{836}\)

\(A=\frac{1}{1.4.7}+\frac{1}{4.7.10}+...+\frac{1}{54.57.60}\)

\(\Rightarrow6A=\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)

\(=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.47}-\frac{1}{57.60}\)

\(=\frac{1}{4}-\frac{1}{3420}=\frac{855}{3420}-\frac{1}{3420}=\frac{427}{1710}\)

\(\Rightarrow A=\frac{427}{1710}:6=\frac{427}{1710}.\frac{1}{6}=\frac{427}{10260}\)

Nhận thấy: 

\(\frac{6}{1.4.7}=\frac{1}{1.4}-\frac{1}{4.7}\)

...............

\(\frac{6}{54.57.60}=\frac{1}{54.57}-\frac{1}{57.60}\)

=> ta phải nhân A vói 6 

=> 6A = 

\(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.57}-\frac{1}{57.60}=\frac{1}{4}-\frac{1}{57.60}=\frac{427}{1710}\)

=> A = 427/1710 : 6 =427/10260

\(S=\dfrac{1}{1.4.7}+\dfrac{1}{4.7.10}+...+\dfrac{1}{22.25.28}\)

\(=\dfrac{1}{6}\left(\dfrac{6}{1.4.7}+\dfrac{6}{4.7.10}+...+\dfrac{6}{22.25.28}\right)\)

\(=\dfrac{1}{6}\left(\dfrac{1}{1.4}-\dfrac{1}{4.7}+\dfrac{1}{4.7}-\dfrac{1}{7.10}+...+\dfrac{1}{22.25}-\dfrac{1}{25.28}\right)\)

\(=\dfrac{1}{6}\left(\dfrac{1}{4}-\dfrac{1}{25.28}\right)\)

\(=\dfrac{1}{24}-\dfrac{1}{6.25.28}\)

Vậy...

\(S=\dfrac{1}{1.4.7}+\dfrac{1}{4.7.10}+\dfrac{1}{7.10.13}+...+\dfrac{1}{22.25.28}\)

\(\Rightarrow6S=\dfrac{6}{1.4.7}+\dfrac{6}{4.7.10}+\dfrac{6}{7.10.13}+...+\dfrac{6}{22.25.28}\)

\(\Rightarrow6S=\dfrac{1}{1.4}-\dfrac{1}{4.7}+\dfrac{1}{4.7}-\dfrac{1}{7.10}+...+\dfrac{1}{22.25}-\dfrac{1}{25.28}\)

\(\Rightarrow6S=\dfrac{1}{1.4}-\dfrac{1}{25.28}\)

\(\Rightarrow6S=\dfrac{1}{4}-\dfrac{1}{700}=\dfrac{87}{350}\)

\(\Rightarrow S=\dfrac{29}{700}\)

Chúc bạn học tốt!!!