\(A=\dfrac{1}{3}\left(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+...+\dfrac{1}{123}-\dfrac{1}{126}\right)\)

\(A=\dfrac{1}{3}\left(\dfrac{1}{3}-\dfrac{1}{126}\right)=\dfrac{1}{3}\cdot\dfrac{41}{126}=\dfrac{41}{378}\)

So \(A=\dfrac{41}{378}\)

We see that:

\(\dfrac{1}{3.6}\)=\(\dfrac{1}{3}\). \(\dfrac{3}{3.6}\)=\(\dfrac{1}{3}\). \(\dfrac{1}{3}\)- \(\dfrac{1}{6}\)

=>A=\(\dfrac{1}{3}\). \(\dfrac{1}{3}\)- \(\dfrac{1}{6}\)+ \(\dfrac{1}{6}\)- \(\dfrac{1}{9}\)+...+\(\dfrac{1}{123}\)- \(\dfrac{1}{126}\)

       =\(\dfrac{1}{3}\). \(\dfrac{1}{3}\)- \(\dfrac{1}{126}\)

       =\(\dfrac{41}{378}\)