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A = \(\dfrac{2}{9}\) + \(\dfrac{2}{45}\) + \(\dfrac{2}{105}\)+ \(\dfrac{2}{189}\)+....+ \(\dfrac{2}{1725}\)

A = \(\dfrac{2}{3\times3}\)+ \(\dfrac{2}{3\times15}\)+ \(\dfrac{2}{3\times35}\)+\(\dfrac{2}{3\times63}\)+...+ \(\dfrac{2}{3\times575}\)

A = \(\dfrac{1}{3}\) \(\times\) ( \(\dfrac{2}{3}\)  + \(\dfrac{2}{15}\) + \(\dfrac{2}{35}\) + \(\dfrac{2}{63}\)+ ...+ \(\dfrac{2}{575}\) )

A = \(\dfrac{1}{3}\) \(\times\) ( \(\dfrac{2}{1\times3}\) + \(\dfrac{2}{3\times5}\)+ \(\dfrac{2}{5\times7}\)+ \(\dfrac{2}{7\times9}\)+...+ \(\dfrac{2}{23\times25}\))

A  = \(\dfrac{1}{3}\) \(\times\) ( \(\dfrac{1}{1}\) - \(\dfrac{1}{3}\) + \(\dfrac{1}{3}\) - \(\dfrac{1}{5}\) + \(\dfrac{1}{5}\) - \(\dfrac{1}{7}\) + \(\dfrac{1}{7}\) - \(\dfrac{1}{9}\)+...+ \(\dfrac{1}{23}\) - \(\dfrac{1}{25}\))

A =  \(\dfrac{1}{3}\) \(\times\) ( \(\dfrac{1}{1}\) - \(\dfrac{1}{25}\))

A = \(\dfrac{1}{3}\) \(\times\)  \(\dfrac{24}{25}\)

A = \(\dfrac{8}{25}\)

A = 8/25

\(=\dfrac{2}{1x3x3}+\dfrac{2}{3x5x3}+\dfrac{2}{5x7x3}+\dfrac{2}{7x9x3}+...+\dfrac{2}{23x25x3}\)

\(=(\dfrac{2}{1x3}+\dfrac{2}{3x5}+\dfrac{2}{5x7}+\dfrac{2}{7x9}+...+\dfrac{2}{23x25})x\dfrac{1}{3}\)

\(=(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{23}-\dfrac{1}{25})x\dfrac{1}{3}\)

\(=(\dfrac{1}{1}-\dfrac{1}{25})x\dfrac{1}{3}\)

\(=\dfrac{24}{25}x\dfrac{1}{3}\)

\(=\dfrac{8}{25}\)