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Ta có \(A=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+\frac{1}{7.10}-...+\frac{1}{16.19}-\frac{1}{19.22}\)
\(=\frac{1}{4}-\frac{1}{418}=\frac{207}{836}\)
\(A=\frac{6}{1\cdot4\cdot7}+\frac{6}{4\cdot7\cdot10}+\frac{6}{7\cdot10\cdot13}+...+\frac{6}{16\cdot19\cdot22}\)
\(A=\frac{1}{1\cdot4}-\frac{1}{4\cdot7}+\frac{1}{4\cdot7}-\frac{1}{7\cdot10}+...+\frac{1}{16\cdot19}-\frac{1}{19\cdot22}\)
\(A=\frac{1}{4}-\frac{1}{19\cdot22}=\frac{207}{836}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)
\(=3.\left(\frac{1}{1.4}-\frac{1}{4.7}\right)+3.\left(\frac{1}{4.7}-\frac{1}{7.10}\right)+...+3.\left(\frac{1}{54.57}-\frac{1}{57.60}\right)\)
\(=3\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...-\frac{1}{54.57}+\frac{1}{54.57}-\frac{1}{57.60}\right)\)
\(=3\left(\frac{1}{1.4}-\frac{1}{57.60}\right)\)
\(=3\left(\frac{1}{4}-\frac{1}{3420}\right)\)
\(=3\left(\frac{855}{3420}-\frac{1}{3420}\right)\)
\(=3.\frac{427}{1710}\)
\(=\frac{427}{570}\)
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\(A=\frac{1}{1.4.7}+\frac{1}{4.7.10}+...+\frac{1}{54.57.60}\)
\(\Rightarrow6A=\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)
\(=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.47}-\frac{1}{57.60}\)
\(=\frac{1}{4}-\frac{1}{3420}=\frac{855}{3420}-\frac{1}{3420}=\frac{427}{1710}\)
\(\Rightarrow A=\frac{427}{1710}:6=\frac{427}{1710}.\frac{1}{6}=\frac{427}{10260}\)
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
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\(S=\dfrac{1}{1.4.7}+\dfrac{1}{4.7.10}+...+\dfrac{1}{22.25.28}\)
\(=\dfrac{1}{6}\left(\dfrac{6}{1.4.7}+\dfrac{6}{4.7.10}+...+\dfrac{6}{22.25.28}\right)\)
\(=\dfrac{1}{6}\left(\dfrac{1}{1.4}-\dfrac{1}{4.7}+\dfrac{1}{4.7}-\dfrac{1}{7.10}+...+\dfrac{1}{22.25}-\dfrac{1}{25.28}\right)\)
\(=\dfrac{1}{6}\left(\dfrac{1}{4}-\dfrac{1}{25.28}\right)\)
\(=\dfrac{1}{24}-\dfrac{1}{6.25.28}\)
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!!!