=>2A=4/2*6+4/6*10+4/10*14+...+4/42*46
=>2A=1/2-1/6+1/6-1/10+...+1/42-1/46=1/2-1/46=22/46
=>A=11/46
\(A=\dfrac{1}{6}+\dfrac{1}{30}+\dfrac{1}{70}+\dfrac{1}{126}+...+\dfrac{1}{966}\)
\(\Rightarrow\dfrac{4}{2}A=\dfrac{4}{2}\cdot\left(\dfrac{1}{6}+\dfrac{1}{30}+\dfrac{1}{70}+...+\dfrac{1}{966}\right)\)
\(\Rightarrow2A=\dfrac{4}{2\times6}+\dfrac{4}{6\times10}+\dfrac{4}{10\times14}+\dfrac{4}{14\times18}+...+\dfrac{4}{42\times46}\)
\(\Rightarrow2A=\dfrac{1}{2}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{18}+...+\dfrac{1}{42}-\dfrac{1}{46}\)
\(\Rightarrow2A=\dfrac{1}{2}-\dfrac{1}{46}\)
\(\Rightarrow2A=\dfrac{11}{23}\)
\(\Rightarrow A=\dfrac{11}{23}:2\)
\(\Rightarrow A=\dfrac{11}{46}\)
