Tính nhanh tổng a = 1 phần 20 cộng 1 phần 30 + 1 phần 42 + 1,56 + trong dung Do chấm + 1 phần 990
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+...+\dfrac{1}{110}\)
= \(\dfrac{1}{3\times4}+\dfrac{1}{4\times5}+\dfrac{1}{5\times6}+...+\dfrac{1}{10\times11}\)
= \(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{10}-\dfrac{1}{11}\)
= \(\dfrac{1}{3}-\dfrac{1}{11}=\dfrac{8}{33}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
#)Giải :
Bài 1 :
\(A=\frac{1}{5}+\frac{1}{10}+\frac{1}{20}+...+\frac{1}{1280}\)
\(\Rightarrow A\times2=\frac{2}{5}-\left(\frac{1}{5}+\frac{1}{10}+\frac{1}{20}+...+\frac{1}{1280}\right)-\frac{1}{1280}\)
\(\Rightarrow A\times2=\frac{2}{5}-A-\frac{1}{1280}\)
\(\Rightarrow A\times2+A=\frac{2}{5}-\frac{1}{1280}\)
\(\Rightarrow A=\frac{2}{5}-\frac{1}{1280}\)
\(\Rightarrow A=\frac{511}{1280}\)
#)Giải :
Bài 2 :
\(B=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{59049}\)
\(B=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{10}}\)
\(3B=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^9}\)
\(3B-B=\left(1+\frac{1}{3}+\frac{1}{3^2}...+\frac{1}{3^9}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{10}}\right)\)
\(2B=1-\frac{1}{3^{10}}\)
\(B=\frac{1-\frac{1}{3^{10}}}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{18.19.20}\)
\(2A=\dfrac{3-1}{1.2.3}+\dfrac{4-2}{2.3.4}+\dfrac{5-3}{3.4.5}+...+\dfrac{20-18}{18.19.20}=\)
\(=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+\dfrac{1}{3.4}-\dfrac{1}{4.5}+...+\dfrac{1}{18.19}-\dfrac{1}{19.20}=\dfrac{1}{2}-\dfrac{1}{19.20}\)
\(\Rightarrow A=\left(\dfrac{1}{2}-\dfrac{1}{19.20}\right):2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
A=1/4*5 + 1/5*6 + 1/6*7 +.....+1/99*100
A=1/4-1/5+1/5-1/6+1/6-1/7+...+1/99-1/100
A=1/4-1/100
A=25/100-1/100
A=6/25
A = 1/20 + 1/30 + 1/42 + 1/56 + .........+1/990
A = 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8 + ...........+ 1/99.100
A = 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + ....... + 1/99 - 1/100
A = 1/4 - ( -1/5 + 1/5 ) - ( -1/6 + 1/6 ) - ( -1/7 + 1/7 ) - ...........- ( - 1/99 + 1/99 ) - 1/100
A = 1/4 - 0 - 0 - 0 - ........... - 0 - 1/100
A = 1/4 - 1/100
A = 25/100 - 1/100
A = 24/100
A = 6/25
![](https://rs.olm.vn/images/avt/0.png?1311)
\(F=\frac{1}{18}+\frac{1}{54}+\frac{1}{108}+...+\frac{1}{990}\)
\(F=\frac{1}{9}\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{110}\right)\)
\(F=\frac{1}{9}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}\right)\)
\(F=\frac{1}{9}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{10}-\frac{1}{11}\right)\)
\(F=\frac{1}{9}\left(1-\frac{1}{11}\right)\)
\(F=\frac{1}{9}.\frac{10}{11}=\frac{10}{99}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
bài 5 ta tính tổng cả ba số đi rồi quy đồng tử số là ra có gì đâu
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+...+\frac{1}{990}\)
\(=\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{30.31}\)
\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+....+\frac{1}{30}-\frac{1}{31}\)
\(=\frac{1}{4}-\frac{1}{31}\)
\(=\frac{27}{124}\)