Question

E=-1/3.(1+2+3)-1/4.(1+2+3+4)-...-1/50.(1+2+3+4+...+50)

Answer 1

\(E=-\dfrac{1}{3}\cdot\left(1+2+3\right)-\dfrac{1}{4}\left(1+2+3+4\right)-...-\dfrac{1}{50}\left(1+2+3+...+50\right)\)

\(=\dfrac{-1}{3}\cdot\dfrac{3\cdot4}{2}-\dfrac{1}{4}\cdot\dfrac{4\cdot5}{2}-...-\dfrac{1}{50}\cdot\dfrac{50\cdot51}{2}\)

\(=\dfrac{-4}{2}-\dfrac{5}{2}-...-\dfrac{51}{2}\)

\(=\dfrac{-\left(4+5+...+51\right)}{2}\)

\(=\dfrac{-\left(51+4\right)\cdot\dfrac{48}{2}}{2}=-\dfrac{1320}{2}=-660\)