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S=-4/1.4/5-4/5.4/9-4/9.4/13-...-4/2013.4/2017
=-4^2/1.5-4^2/5.9-4^2/9.13-...-4^2/2013.2017
=-4(4/1.5+4/5.9-4/9.13-...-4/2013.2017)
=-4(1-1/5+1/5-1/9+1/9-1/13-...-1/2013-1/2017)
=-4(1-1/2017)
=-4.2016/2017=-8064/2017
Ta có : S= - 4/1 .4/5 - 4/5 .4/9-...- 4/2013 .4/2017
= -4 .( 4/1.5+4/5.9+...+4/2013.2017 )
= -4 . (1-1/5+1/5-1/9+...+1/2013-1/2017 )
= -4 . (1-1/2017)
= -4 .2016/2017
= -8064/2017
=> 2017S= -8064/2017 .2017 = -8064
1. \(\frac{5}{9}.\frac{7}{13}+\frac{5}{9}.\frac{9}{13}-\frac{5}{9}.\frac{3}{13}\)
= \(\frac{5}{9}\) .(\(\frac{7}{13}+\frac{9}{13}-\frac{3}{13}\) )
= \(\frac{5}{9}\) . 1 = \(\frac{5}{9}\)
\(\left(-1\frac{1}{6}\right)\left(\frac{1-\frac{3}{5}+\frac{3}{11}-\frac{3}{13}}{\frac{1}{3}-\frac{1}{5}+\frac{1}{11}-\frac{1}{13}}\right)\left(\frac{4-\frac{4}{17}+\frac{4}{19}-\frac{4}{2013}}{5-\frac{5}{7}+\frac{5}{19}-\frac{5}{2013}}\right)\)
\(=-\frac{7}{6}.\left(\frac{3\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{11}-\frac{1}{13}\right)}{\frac{1}{3}-\frac{1}{5}+\frac{1}{11}-\frac{1}{13}}\right):\left(\frac{4.\left(1-\frac{1}{7}+\frac{1}{19}-\frac{1}{2013}\right)}{5.\left(1-\frac{1}{7}+\frac{1}{19}-\frac{1}{2013}\right)}\right)\)
\(=-\frac{7}{6}.3:\frac{4}{5}=-\frac{7}{2}.\frac{5}{4}=-\frac{35}{8}\)