Question

1/2+(1/2)^2+(1/2)^3+(1/2)^4+....+(1/2)^20

 

Answer 1

\(S=\frac{1}{2}+\left(\frac{1}{2}\right)^2+...+\left(\frac{1}{2}\right)^{20}\)

\(\Rightarrow\frac{1}{2}S=\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{21}\)

\(\Rightarrow\frac{1}{2}S-S=\left(\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{21}\right)-\left(\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)^2+...+\left(\frac{1}{2}\right)^{20}\right)\)

\(\Rightarrow-\frac{1}{2}S=\left(\frac{1}{2}\right)^{21}-\frac{1}{2}\)

\(\Rightarrow S=\left(\left(\frac{1}{2}\right)^{21}-\frac{1}{2}\right):\frac{-1}{2}\)