\(A=1+\left(\dfrac{1}{2}+\dfrac{1}{3}\right)+\left(\dfrac{1}{4}+...+\dfrac{1}{7}\right)+...+\left(\dfrac{1}{2^{99}}+\dfrac{1}{2^{99}+1}+...+\dfrac{1}{2^{100}-1}\right)\)
\(A< 1+\left(\dfrac{1}{2}+\dfrac{1}{2}\right)+\left(\dfrac{1}{4}+...+\dfrac{1}{4}\right)+...+\left(\dfrac{1}{2^{99}}+\dfrac{1}{2^{99}}+...+\dfrac{1}{2^{99}}\right)\)
\(A< 1+2.\dfrac{1}{2}+4.\dfrac{1}{4}+...+2^{99}.\dfrac{1}{2^{99}}\)
\(A< 1+1+1+...+1=100\)
\(A=1+\dfrac{1}{2}+\left(\dfrac{1}{3}+\dfrac{1}{4}\right)+\left(\dfrac{1}{5}+...+\dfrac{1}{8}\right)+...\left(\dfrac{1}{2^{99}+1}+...+\dfrac{1}{2^{100}}\right)-\dfrac{1}{2^{100}}\)
\(A>1+\dfrac{1}{2}++\left(\dfrac{1}{4}+\dfrac{1}{4}\right)+\left(\dfrac{1}{8}+...+\dfrac{1}{8}\right)+...+\left(\dfrac{1}{2^{100}}+...+\dfrac{1}{2^{100}}\right)-\dfrac{1}{2^{100}}\)
\(A>1+\dfrac{1}{2}+2.\dfrac{1}{4}+4.\dfrac{1}{8}+...+2^{99}.\dfrac{1}{2^{100}}-\dfrac{1}{2^{100}}\)
\(A>1+\dfrac{1}{2}+\dfrac{1}{2}+...+\dfrac{1}{2}-\dfrac{1}{2^{100}}=50+1-\dfrac{1}{2^{100}}>50\)
