Question

(-1/7)0+(-1/7)1+(-1/7)2+......+(-1/7)2007

Answer 1

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

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

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

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