Question

{-1/7}^0 + {-1/7}^1 + { -1/7 } ^2 + ... + { -1/7}^2007

Answer 1

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

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

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

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