Question

s= \(\dfrac{1}{5x9}\)+\(\dfrac{1}{9x13}\)+....+\(\dfrac{1}{41x45}\)

Answer 1

\(S=\dfrac{1}{5\times9}+\dfrac{1}{9\times13}+...+\dfrac{1}{41\times45}\)

\(\Rightarrow4S=\dfrac{4}{5\times9}+\dfrac{4}{9\times13}+...+\dfrac{4}{41\times45}\)

\(\Rightarrow4S=\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{41}-\dfrac{1}{45}\)

\(\Rightarrow4S=\dfrac{1}{5}-\dfrac{1}{45}\)

\(\Rightarrow4S=\dfrac{8}{45}\)

\(\Rightarrow S=\dfrac{2}{45}\)

Answer 2

\(=\dfrac{1}{4}\left(\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{41\cdot45}\right)\)

\(=\dfrac{1}{4}\left(\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{41}-\dfrac{1}{45}\right)\)

\(=\dfrac{1}{4}\cdot\dfrac{8}{45}=\dfrac{2}{45}\)