Welcome 2016! Part 20

Algebra Level 4

\[\large \dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\cdots+\dfrac{1}{1+2+\cdots+20}\]

If the value of the expression above is equal to \( \dfrac mn\), where \(m\) and \(n\) are coprime positive integers, find the value of \(m+n\).