\(B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^{3.}}+.............+\frac{1}{2^{100}}\)
\(2B=1+\frac{1}{2}+\frac{1}{2^2}+.................+\frac{1}{2^{99}}\)
\(2B-B=1-\frac{1}{2^{100}}\)
\(B=1-\frac{1}{2^{100}}\)
\( C=\frac{1}{2}-\frac{1}{2^2}+.................+\frac{1}{2^{99}}-\frac{1}{2^{100}}\)
\(2 C=1-\frac{1}{2}+......................+\frac{1}{2^{98}}-\frac{1}{2^{99}}\)
\(2 C+C=1-\frac{1}{2^{100}}\)
\(C=\left(1-\frac{1}{2^{100}}\right):3\)