:) 

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

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

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

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

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

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

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

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

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

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

S=a^0+a^1+a^2+....+a^2007 (1) <=>a.S=a^1+a^2+a^3+....+a^2007+a^2008 (2) lấy (2) trừ (1) ta được: a.S-S=a^2008-a^0=a^2008-1 <=>S=(a^2008-1)/(a-1) với a=-1/7 ta có: S= (-1/7)^0 + (-1/7)^1+(-1/7)^2 +...+ (-1/7)^2007 =[(-1/7)^2008 -1]/(-1/7 -1)

a) \(S=\left(-\frac{1}{7}\right)^0+\left(-\frac{1}{7}\right)^1+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2007}\)

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

=> 7S = \(7+\left(-1\right)+\left(-\frac{1}{7}\right)+...+\left(-\frac{1}{7}\right)^{2006}\)

Lấy 7S trừ S ta có : 

7S - S = \(7+\left(-1\right)+\left(-\frac{1}{7}\right)+...+\left(-\frac{1}{7}\right)^{2006}-\left[1+\left(-\frac{1}{7}\right)+\left(-\frac{1}{7}\right)^2+...+\left(-\frac{1}{7}\right)^{2007}\right]\)

6S = \(7-1-1+\left(\frac{1}{7}\right)^{2007}=5+\left(\frac{1}{7}\right)^{2007}\Rightarrow S=\frac{5+\left(\frac{1}{7}\right)^{2007}}{6}\)