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S=(-1/7)0+(-1/7)1+...+(-1/7)2007
-1/7.S=(-1/7)1+(-1/7)2+...+(-1/7)2008
-1/7.S-S=[(-1/7)1+(-1/7)2+...+(-1/7)2008]-[(-1/7)0+(-1/7)1+...+(-1/7)2007]
-8/7.S=(-1/7)2008-(-1/7)0
-8/7.S=(1/7)2008-1
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\(S=\left(-\frac{1}{7}\right)^0+\left(-\frac{1}{7}\right)^1+...+\left(-\frac{1}{7}\right)^{2007}\)
\(-\frac{1}{7}S=\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2008}\)
\(-\frac{1}{7}S-S=\left(-\frac{1}{7}\right)^0+\left(-\frac{1}{7}\right)^{2008}\)
\(-\frac{8}{7}S=1+\frac{\left(-1\right)^{2008}}{7^{2008}}=1+\frac{1}{7^{2008}}=\frac{7^{2008}+1}{7^{2008}}\)
\(S=\frac{7^{2008}+1}{7^{2008}}:\left(-\frac{8}{7}\right)\)
HOK TOT
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\(S=\left(-\dfrac{1}{7}\right)^0+\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+...+\left(-\dfrac{1}{7}\right)^{2017}\\ S=\dfrac{\left(-1\right)^0}{7^0}+\dfrac{\left(-1\right)^1}{7^1}+\dfrac{\left(-1\right)^2}{7^2}+...+\dfrac{\left(-1\right)^{2017}}{7^{2017}}\\ S=\dfrac{1}{7^0}+\dfrac{-1}{7^1}+\dfrac{1}{7^2}+...+\dfrac{-1}{7^{2017}}\\ -7S=\dfrac{-7}{7^0}+\dfrac{7}{7^1}+\dfrac{-7}{7^2}+...+\dfrac{7}{7^{2017}}\\ -7S=\left(-7\right)+\dfrac{1}{7^0}+\dfrac{-1}{7^1}+...+\dfrac{1}{7^{2016}}\\ -7S-S=\left[\left(-7\right)+\dfrac{1}{7^0}+\dfrac{-1}{7^1}+...+\dfrac{1}{7^{2016}}\right]+\left(\dfrac{1}{7^0}+\dfrac{-1}{7^1}+\dfrac{1}{7^2}+...+\dfrac{-1}{7^{2017}}\right)\\ -8S=\left(-7\right)+\dfrac{-1}{2017}\\ -8S=-\left(7+\dfrac{1}{2017}\right)\\ 8S=7+\dfrac{1}{2017}\\ S=\dfrac{7+\dfrac{1}{2017}}{8}\)
Vậy ...
\(S=\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+...........+\left(\frac{-1}{7}\right)^{2007}\)
\(\Rightarrow\frac{-1}{7}S=\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+........+\left(\frac{-1}{7}\right)^{2008}\)
\(\Rightarrow\frac{-1}{7}S-S=\frac{-8}{7}S=\left(\frac{-1}{7}\right)^{2008}-\left(\frac{-1}{7}\right)^0\)
\(\Rightarrow S=\frac{\left(\frac{-1}{7}\right)^{2008}-1}{\frac{-8}{7}}\)