Đặt \(A=\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+...+\left(\frac{-1}{7}\right)^{2007}\)
\(\frac{-1}{7}.A=\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+\left(\frac{-1}{7}\right)^3+...+\left(\frac{-1}{7}\right)^{2008}\)
\(A-\frac{-1}{7}.A=\left[\left(\frac{-1}{7}\right)^0+\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+...+\left(\frac{-1}{7}\right)^{2007}\right]-\left[\left(\frac{-1}{7}\right)^1+\left(\frac{-1}{7}\right)^2+\left(\frac{-1}{7}\right)^3+...+\left(\frac{-1}{7}\right)^{2008}\right]\)
\(A+\frac{1}{7}.A=\left(\frac{-1}{7}\right)^0-\left(\frac{-1}{7}\right)^{2008}\)
\(\frac{8}{7}.A=1-\left(\frac{1}{7}\right)^{2008}\)
\(\frac{8}{7}.A=1-\frac{1}{7^{2008}}\)
\(A=\left(1-\frac{1}{7^{2008}}\right):\frac{8}{7}=\frac{\left(1-\frac{1}{7^{2008}}\right).7}{8}\)
