\(A=\frac{2}{3}+\frac{1}{3}+\frac{1}{15}+\frac{1}{75}+\frac{1}{375}+\frac{1}{1875}\)
\(A=\left(\frac{2}{3}+\frac{1}{3}\right)+\frac{1}{15}+\frac{1}{75}+\frac{1}{375}+\frac{1}{1875}\)
\(A=1+\left(\frac{5}{75}+\frac{1}{75}\right)+\frac{1}{375}+\frac{1}{1875}\)
\(A=1+\frac{6}{75}+\frac{1}{375}+\frac{1}{1875}\)
\(A=1+\left(\frac{150}{1875}+\frac{1}{1875}\right)+\frac{1}{375}\)
\(A=1+\frac{151}{1875}+\frac{1}{375}\)
\(A=1+\left(\frac{151}{1875}+\frac{5}{1875}\right)\)
\(A=1+\frac{156}{1875}\)
\(A=\frac{1875}{1875}+\frac{156}{1875}\)
\(A=\frac{677}{625}\)
\(A=\frac{2}{3}+\frac{1}{3}+\frac{1}{15}+\frac{1}{75}+\frac{1}{375}+\frac{1}{1875}\)
\(\Leftrightarrow A=\left(\frac{2}{3}+\frac{1}{3}\right)+\left(\frac{1}{15}+\frac{1}{75}\right)+\left(\frac{1}{375}+\frac{1}{1875}\right)\)
\(\Leftrightarrow A=1+\left(\frac{5}{75}+\frac{1}{75}\right)+\left(\frac{5}{1875}+\frac{1}{1875}\right)\)
\(\Leftrightarrow A=1+\frac{6}{75}+\frac{6}{1875}\)
\(\Leftrightarrow A=1+\frac{2}{25}+\frac{2}{625}\)
\(\Leftrightarrow A=1+\left(\frac{2}{25}+\frac{2}{625}\right)\)
\(\Leftrightarrow A=1+\left(\frac{50}{625}+\frac{2}{625}\right)\)
\(\Leftrightarrow A=1+\frac{52}{625}\)
\(\Leftrightarrow A=\frac{677}{625}\)