\(\dfrac{1}{15}+\dfrac{1}{35}+\dfrac{1}{63}+\dfrac{1}{99}+\dfrac{1}{143}\\ =\dfrac{1}{3\times5}+\dfrac{1}{5\times7}+\dfrac{1}{7\times9}+\dfrac{1}{9\times11}+\dfrac{1}{11\times13}\\ =\left(\dfrac{2}{3\times5}+\dfrac{2}{5\times7}+\dfrac{2}{7\times9}+\dfrac{2}{9\times11}+\dfrac{2}{11\times13}\right).\dfrac{1}{2}\\ =\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{13}\right).\dfrac{1}{2}\\ =\left(\dfrac{1}{3}-\dfrac{1}{13}\right).\dfrac{1}{2}\\ =\dfrac{10}{39}.\dfrac{1}{2}\\ =\dfrac{5}{39}\)
115+135+163+199+1143=13×5+15×7+17×9+19×11+111×13=(23×5+25×7+27×9+29×11+211×13).12=(13−15+15−17+17−19+19−111+111−113).12=(13−113).12=1039.12=539
