\(A=\dfrac{3}{2}-\dfrac{5}{6}+\dfrac{7}{12}-\dfrac{9}{20}+.....+\dfrac{23}{132}\)
\(A=\dfrac{3}{1.2}-\dfrac{5}{2.3}+\dfrac{7}{3.4}-\dfrac{9}{4.5}+.....+\dfrac{23}{11.12}\)
\(A=3-\dfrac{3}{2}-\dfrac{5}{2}+\dfrac{5}{3}+\dfrac{7}{3}-\dfrac{7}{4}-\dfrac{9}{4}+\dfrac{9}{5}+...+\dfrac{23}{11}-\dfrac{23}{12}\)
\(A=3-\left(\dfrac{3}{2}+\dfrac{5}{2}\right)+\left(\dfrac{5}{3}+\dfrac{7}{3}\right)-\left(\dfrac{7}{4}+\dfrac{9}{4}\right)+...-\dfrac{23}{12}\)
\(A=3-4+4-4+4-4+4-4+4-4+4-\dfrac{23}{12}\)
\(A=3-\dfrac{23}{12}=\dfrac{36}{12}-\dfrac{23}{12}=\dfrac{13}{12}\)
\(A=\dfrac{41580-23100+16170-12474+10164-8580+7425-6545+5852-5292+4830}{27720}\)
\(A=\dfrac{30030}{27720}=\dfrac{13}{12}\)
