A = \(\dfrac{1}{1+2}\) + \(\dfrac{1}{1+2+3}\) +...+ \(\dfrac{1}{1+2+3+...+9}\)
A = \(\dfrac{1}{\left(2+1\right)\times2:2}\)+ \(\dfrac{1}{\left(1+3\right)\times3:2}\)+...+ \(\dfrac{1}{\left(1+9\right)\times9:2}\)
A = \(\dfrac{2}{2\times\left(1+2\right)}\)+ \(\dfrac{2}{3\times\left(1+4\right)}\)+...+ \(\dfrac{2}{9\times\left(1+9\right)}\)
A = \(\dfrac{2}{2\times3}\) + \(\dfrac{2}{3\times4}\)+....+ \(\dfrac{2}{9\times10}\)
A = 2 \(\times\)( \(\dfrac{1}{2\times3}\) + \(\dfrac{1}{3\times4}\)+...+ \(\dfrac{1}{9\times10}\))
A = 2 \(\times\) ( \(\dfrac{1}{2}\) - \(\dfrac{1}{3}\) + \(\dfrac{1}{3}\) - \(\dfrac{1}{4}\)+...+ \(\dfrac{1}{9}\) - \(\dfrac{1}{10}\))
A = 2 \(\times\)( \(\dfrac{1}{2}\) - \(\dfrac{1}{10}\))
A = 2 \(\times\) \(\dfrac{2}{5}\)
A = \(\dfrac{4}{5}\)
