\(\dfrac{3}{4}.\dfrac{8}{9}.\dfrac{15}{16}....\dfrac{624}{625}\)
\(=\dfrac{3.8.15...624}{4.9.16...625}\)
\(=\dfrac{\left(1.3\right)\left(2.4\right)\left(3.5\right)...\left(24.26\right)}{\left(2.2\right)\left(3.3\right)\left(4.4\right)...\left(25.25\right)}\)
\(=\dfrac{\left(1.2.3...24\right)\left(3.4.5...26\right)}{\left(2.3.4...25\right).\left(2.3.4...25\right)}\)
\(=\dfrac{1.26}{25.2}\)
\(=\dfrac{13}{25}\)
\(\dfrac{3}{4}\cdot\dfrac{8}{9}\cdot\dfrac{15}{16}\cdot\cdot\cdot\dfrac{624}{625}\)
\(=\dfrac{1.3}{2.2}\cdot\dfrac{2.4}{3.3}\cdot\dfrac{3.5}{4.4}\cdot\cdot\cdot\dfrac{24.26}{25.25}\)
\(=\dfrac{1.2.3....24}{2.3.4....25}\cdot\dfrac{3.4.5....26}{2.3.4...25}\)
\(=\dfrac{1}{25}\cdot\dfrac{26}{2}=\dfrac{13}{25}\)
\(\dfrac{3}{4}\cdot\dfrac{8}{9}\cdot\dfrac{15}{16}\cdot....\cdot\dfrac{624}{625}=\dfrac{3}{2\cdot2}\cdot\dfrac{2\cdot4}{3\cdot3}\cdot\dfrac{3\cdot5}{4\cdot4}\cdot...\cdot\dfrac{9\cdot11}{10\cdot10}=\dfrac{11}{2\cdot10}=\dfrac{11}{20}\)