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\(=\frac{1}{2}.\left(\frac{1}{2011}-\frac{1}{2009}+\frac{1}{2009}-....+\frac{1}{3}-1\right)\)
\(=\frac{1}{2}.\left(\frac{1}{2011}-1\right)\)
\(=\frac{1}{2}.\frac{-2012}{2011}=\frac{-1006}{2011}\)
S = \(\frac{1}{3x5}+\frac{1}{5x7}+\frac{1}{7x9}+...+\frac{1}{17x19}\)
2S = \(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\)\(\frac{1}{17}-\frac{1}{19}\)
2S = \(\frac{1}{3}-\frac{1}{19}\)
2S = \(\frac{16}{57}\)
S = \(\frac{16}{57}\times\frac{1}{2}\)
S = \(\frac{8}{57}\)
\(S=\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}+\frac{1}{255}+\frac{1}{323}\)
\(S=\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+\frac{1}{9\cdot11}+\frac{1}{11\cdot13}+\frac{1}{13\cdot15}+\frac{1}{15\cdot17}+\frac{1}{17\cdot19}\)
\(2S=\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+...+\frac{2}{15\cdot17}+\frac{2}{17\cdot19}\)
\(2S=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{15}-\frac{1}{17}+\frac{1}{17}-\frac{1}{19}\)
\(2S=\frac{1}{3}-\frac{1}{19}\)
\(2S=\frac{19}{57}-\frac{3}{57}\)
\(2S=\frac{16}{57}\)
\(S=\frac{16}{57}:2\)
\(S=\frac{16}{57}\cdot\frac{1}{2}\)
\(S=\frac{8}{57}\)