Question

1/1*2*3+1/2*3*4+1/3*4*5+...+1/50*51*52

Answer 1

\(=\dfrac{1}{2}\cdot\left(\dfrac{2}{1\cdot2\cdot3}+\dfrac{2}{2\cdot3\cdot4}+...+\dfrac{2}{50\cdot51\cdot52}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{50\cdot51}-\dfrac{1}{51\cdot52}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{51\cdot52}\right)\)

\(=\dfrac{1}{2}\cdot\dfrac{1325}{2652}=\dfrac{1325}{5304}\)