S = \(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{100}}\)
2S = \(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}\)
2S - S = \(1-\frac{1}{2^{100}}\)
=> S = \(1-\frac{1}{2^{100}}\)
\(2S=1+\frac{1}{2}+\frac{1}{2^2}+.......+\frac{1}{2^{99}}\)
\(2S-S=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{100}}\right)\)
\(S=1-\frac{1}{2^{100}}\)