Question

    5                5                   5       

_______ +_______+.....+_______

  3×4           4×5              49×50

Answer 1

\(\dfrac{5}{3\times4}+\dfrac{5}{4\times5}+...+\dfrac{5}{49\times50}\)

\(=5\times\left(\dfrac{1}{3\times4}+\dfrac{1}{4\times5}+...+\dfrac{1}{49\times50}\right)\)

\(=5\times\left(\dfrac{4-3}{3\times4}+\dfrac{5-4}{4\times5}+...+\dfrac{50-49}{49\times50}\right)\)

\(=5\times\left(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{49}-\dfrac{1}{50}\right)\)

\(=5\times\left(\dfrac{1}{3}-\dfrac{1}{50}\right)=5\times\left(\dfrac{50}{150}-\dfrac{3}{150}\right)\)

\(=\dfrac{5\times47}{150}=\dfrac{47}{30}\)