\(Z=\dfrac{3}{3\times5}+\dfrac{3}{5\times7}+\dfrac{3}{7\times9}+...+\dfrac{3}{49\times51}\)
\(=\dfrac{3}{2}\times\left(\dfrac{2}{3\times5}+\dfrac{2}{5\times7}+\dfrac{2}{7\times9}+...+\dfrac{2}{49\times51}\right)\)
\(=\dfrac{3}{2}\times\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{49}-\dfrac{1}{51}\right)\)
\(=\dfrac{3}{2}\times\left(\dfrac{1}{3}-\dfrac{1}{51}\right)\)
\(=\dfrac{3}{2}\times\dfrac{16}{51}=\dfrac{8}{17}\)
\(Z=\dfrac{3}{3x5}+\dfrac{3}{5x7}+\dfrac{3}{7x9}+...+\dfrac{3}{49x51}\\ =\dfrac{3}{2}x\left(\dfrac{2}{3x5}+\dfrac{2}{5x7}+\dfrac{2}{7x9}+...+\dfrac{2}{49x51}\right)\\ =\dfrac{3}{2}x\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{49}-\dfrac{1}{51}\right)\\ =\dfrac{3}{2}x\left(\dfrac{1}{3}-\dfrac{1}{51}\right)\\ =\dfrac{3}{2}x\dfrac{16}{51}=\dfrac{8}{17}\)
\(Z=\dfrac{3}{3\times5}+\dfrac{3}{5\times7}+\dfrac{3}{7\times9}+...+\dfrac{3}{49\times51}\)
\(Z=\dfrac{3}{2}\times\left(\dfrac{1}{3\times5}+\dfrac{1}{5\times7}+\dfrac{1}{7\times9}+...+\dfrac{1}{49\times51}\right)\)
\(Z=\dfrac{3}{2}\times\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7} +...+\dfrac{1}{49}-\dfrac{1}{51}\right)\)
\(Z=\dfrac{3}{2}\times\left(\dfrac{1}{3}-\dfrac{1}{51}\right)\)
\(Z=\dfrac{3}{2}\times\dfrac{16}{51}\)
\(Z=1\times\dfrac{8}{17}\)
\(=>Z=\dfrac{8}{17}\)