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

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

\(S=\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+...........+\left(\frac{-1}{7}\right)^{2007}\)

\(\Rightarrow\frac{-1}{7}S=\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+........+\left(\frac{-1}{7}\right)^{2008}\)

\(\Rightarrow\frac{-1}{7}S-S=\frac{-8}{7}S=\left(\frac{-1}{7}\right)^{2008}-\left(\frac{-1}{7}\right)^0\)

\(\Rightarrow S=\frac{\left(\frac{-1}{7}\right)^{2008}-1}{\frac{-8}{7}}\)

\(S=\left(-\dfrac{1}{7}\right)^0+\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+...+\left(-\dfrac{1}{7}\right)^{2017}\\ S=\dfrac{\left(-1\right)^0}{7^0}+\dfrac{\left(-1\right)^1}{7^1}+\dfrac{\left(-1\right)^2}{7^2}+...+\dfrac{\left(-1\right)^{2017}}{7^{2017}}\\ S=\dfrac{1}{7^0}+\dfrac{-1}{7^1}+\dfrac{1}{7^2}+...+\dfrac{-1}{7^{2017}}\\ -7S=\dfrac{-7}{7^0}+\dfrac{7}{7^1}+\dfrac{-7}{7^2}+...+\dfrac{7}{7^{2017}}\\ -7S=\left(-7\right)+\dfrac{1}{7^0}+\dfrac{-1}{7^1}+...+\dfrac{1}{7^{2016}}\\ -7S-S=\left[\left(-7\right)+\dfrac{1}{7^0}+\dfrac{-1}{7^1}+...+\dfrac{1}{7^{2016}}\right]+\left(\dfrac{1}{7^0}+\dfrac{-1}{7^1}+\dfrac{1}{7^2}+...+\dfrac{-1}{7^{2017}}\right)\\ -8S=\left(-7\right)+\dfrac{-1}{2017}\\ -8S=-\left(7+\dfrac{1}{2017}\right)\\ 8S=7+\dfrac{1}{2017}\\ S=\dfrac{7+\dfrac{1}{2017}}{8}\)

Vậy ...

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S= \(\left(-\dfrac{1}{7}\right)^0+\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+...+\left(-\dfrac{1}{7}\right)^{2017}\)

\(\left(-\dfrac{1}{7}\right)S=\left(-\dfrac{1}{7}\right)\left(-\dfrac{1}{7}+-\dfrac{1^2}{7}+..+-\dfrac{1^{2007}}{7}\right)\)

= \(-\dfrac{1}{7}+-\dfrac{1}{7}^2+....+\dfrac{-1^{2008}}{7}\)

=>\(-\dfrac{1}{7}S-S=\) \(-\dfrac{1}{7}+-\dfrac{1}{7}^2+....+\dfrac{-1^{2008}}{7}\) \(-\)\(\left(1+-\dfrac{1}{7}+-\dfrac{1^2}{7}+...+-\dfrac{1^{2007}}{7}\right)\)

=> \(-\dfrac{1}{7}S=\) \(\dfrac{-1^{2008}}{7}-1\)

=> S= \(\dfrac{-1^{2008}}{7}-1\) : \(\dfrac{-1}{7}\)

S=1+(-1/7)^1+(-1/7)^2+...+(-1/7)^2007

=>7S=7+(-1/7)^1+(1/7)^2+...+(-1/7)^2006

=>(7-1)S=6-(1/7)^2007

=>S=1-(-1/7^2007/6)

1/7S=(-1/7)^1+...+(-1/7)2018

1/7S-S=(-1/7)^1+....+(-1/7)^2018-(-1/7)^0-...-(-1/7)^2017

-6/7S=(-1/7)^2018-1=(-1/7)^2018-1:-6/7

S=

7S = 1+(−1/7)^1+(−1/7)^2+...+(−1/7)^2007

=> 7S = 7+(−1/7)^1+(−1/7)^2+...+(−1/7)^2006

=> 6S = 6-(−1/7)^2007

=> S= 1-(−1/7^2007/6)

sai rùi bạn à bài này mình biết làm rùi

\(A=\left(-\dfrac{1}{7}\right)^0+\left(-\dfrac{1}{7}\right)^1+...+\left(-\dfrac{1}{7}\right)^{2007}\)

\(\Leftrightarrow-\dfrac{1}{7}A=\left(-\dfrac{1}{7}\right)^1+\left(-\dfrac{1}{7}\right)^2+...+\left(-\dfrac{1}{7}\right)^{2008}\)

\(\Leftrightarrow-\dfrac{8}{7}A=\left(-\dfrac{1}{7}\right)^{2008}-1=\dfrac{1}{7^{2008}}-1=\dfrac{1-7^{2008}}{7^{2008}}\)

\(\Leftrightarrow A=\dfrac{1-7^{2008}}{7^{2008}}\cdot\dfrac{-7}{8}=\dfrac{7^{2008}-1}{8\cdot7^{2007}}\)