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Công thức:
Số các số hạng là:
(số cuối-số đầu):khoảng cách+1=số hạng
tổng:(số cuối+số đầu)x số hạng:2=k quả
![](https://rs.olm.vn/images/avt/0.png?1311)
*/ Tổng của 3 số tự nhiên liên tiếp có dạng: a+(a+1)+(a+2)=3a+3=3(a+1) => Luôn chia hết cho 3
*/ 215+424=2.214+2.212=2(214+212) => Luôn chia hết cho 2
*/ \(S1=\frac{2012\left(2012-1\right)}{2}-1-2=2023063\)
*/ \(S2=\frac{2012\left(2012-1\right)}{2}-1=2023065\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{2012}{3^{2012}}\)
\(\Rightarrow3A=1+\frac{2}{3}+\frac{3}{3^2}+\frac{4}{3^3}+...+\frac{2012}{3^{2011}}\)
\(\Rightarrow3A-A=\left(1+\frac{2}{3}+\frac{3}{3^2}+\frac{4}{3^3}+...+\frac{2012}{3^{2011}}\right)-\left(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{2012}{3^{2012}}\right)\)
\(\Rightarrow2A=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+....+\frac{1}{3^{2011}}-\frac{2012}{3^{2012}}\)
\(\Rightarrow6A=3+1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2010}}-\frac{2012}{3^{2011}}\)
\(\Rightarrow6A-2A=\left(3+1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2010}}-\frac{2012}{3^{2011}}\right)-\left(1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+....+\frac{1}{3^{2011}}-\frac{2012}{3^{2012}}\right)\)
\(\Rightarrow4A=3-\frac{2012}{3^{2011}}\)
\(\Rightarrow A=\frac{3-\frac{2012}{3^{2011}}}{4}=\frac{3}{4}-\frac{\frac{2012}{3^{2011}}}{4}=\frac{3}{4}-\frac{2012}{3^{2011}.4}\)
\(\Rightarrow A< \frac{3}{4}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(a,\dfrac{1}{3}-\left(-1\dfrac{2}{5}\right)+\left(-3\dfrac{1}{4}\right)\\ =\dfrac{1}{3}-\left(-\dfrac{7}{5}\right)-\dfrac{13}{4}\\ =\dfrac{1}{3}+\dfrac{7}{5}-\dfrac{13}{4}\\ =\dfrac{20}{3\times20}+\dfrac{7\times12}{5\times12}-\dfrac{13\times15}{4\times15}\\ =\dfrac{20+84-195}{60}\\ =\dfrac{-91}{60}\)
\(b,\dfrac{5}{4}-\left(-3\dfrac{1}{2}\right)-\dfrac{7}{10}\\ =\dfrac{5}{4}+\dfrac{7}{2}-\dfrac{7}{10}\\ =\dfrac{5\times5}{4\times5}+\dfrac{7\times10}{2\times10}-\dfrac{7\times2}{10\times2}\\ =\dfrac{25+70-14}{20}\\ =\dfrac{81}{20}\)
\(c,\dfrac{3}{2}-\left[\left(-\dfrac{4}{7}-\left(\dfrac{1}{2}+\dfrac{5}{8}\right)\right)\right]\\ =\dfrac{3}{2}-\left[-\dfrac{4}{7}-\left(\dfrac{4}{8}+\dfrac{5}{8}\right)\right]\\ =\dfrac{3}{2}-\left(-\dfrac{4}{7}-\dfrac{9}{8}\right)\\ =\dfrac{3}{2}+\dfrac{4}{7}+\dfrac{9}{8}\\ =\dfrac{3\times28}{2\times28}+\dfrac{4\times8}{8\times7}+\dfrac{9\times7}{8\times7}\\ =\dfrac{84+32+63}{56}\\ =\dfrac{179}{56}\)
`@` `\text {Ans}`
`\downarrow`
`a,`
\(\dfrac{1}{3}-\left(-1\dfrac{2}{5}\right)+\left(-3\dfrac{1}{4}\right)\)
`= 1/3+7/5 - 13/4`
`= 26/15 - 13/4`
`= -91/60`
`b,`
\(\dfrac{5}{4}-\left(-3\dfrac{1}{2}\right)-\dfrac{7}{10}\)
`= 5/4+7/2 - 7/10`
`= 1,25 + 3,5 - 0,7`
`= 4,75 - 0,7`
`= 4,05`
`c,`
\(\dfrac{3}{2}-\left[\left(-\dfrac{4}{7}\right)-\left(\dfrac{1}{2}+\dfrac{5}{8}\right)\right]\)
`= 3/2 - [(-4/7) - 9/8]`
`= 3/2 - (-95/56)`
`= 179/56`