\(\dfrac{5}{8}+\dfrac{5}{24}+\dfrac{5}{48}+...+\dfrac{5}{528}\)
\(=\dfrac{5}{2.4}+\dfrac{5}{4.6}+\dfrac{5}{6.8}+...+\dfrac{5}{22.24}\)
\(=\dfrac{5}{2}.\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{22.24}\right)\)
\(=\dfrac{5}{2}.\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-...-\dfrac{1}{22}+\dfrac{1}{22}-\dfrac{1}{24}\right)\)
\(=\dfrac{5}{2}.\left(\dfrac{1}{2}-\dfrac{1}{24}\right)\)
\(=\dfrac{5}{2}.\dfrac{11}{24}\)
\(=\dfrac{55}{48}\).
A=52.4+54.6+56.8+...+522.24=52.4+54.6+56.8+...+522.24
=52.(12−14+14−16+16−...−122+122−124)=52.(12−14+14−16+16−...−122+122−124)
=52.1124=52.1124
