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\(P=\frac{\frac{3}{7}-\frac{3}{13}+\frac{3}{37}}{\frac{5}{7}-\frac{5}{17}+\frac{5}{37}}+\frac{\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}}{\frac{7}{5}-\frac{7}{4}+\frac{7}{3}-\frac{7}{2}}\)
\(=\frac{3\left(\frac{1}{7}-\frac{1}{13}+\frac{1}{37}\right)}{5\left(\frac{1}{7}-\frac{1}{17}+\frac{1}{37}\right)}+\frac{\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}}{-7\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}\right)}\)
\(=\frac{3}{5}+\frac{1}{-7}\)
\(=\frac{16}{35}\)
\(S=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+......+\frac{1}{2011}-\frac{1}{2012}+\frac{1}{2013}\)
\(=\left(1+\frac{1}{3}+......+\frac{1}{2013}\right)-\left(\frac{1}{2}+\frac{1}{4}+.....+\frac{1}{2012}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{2013}\right)-\left(\frac{1}{2}+\frac{1}{4}+....+\frac{1}{2012}\right)-\left(\frac{1}{2}+\frac{1}{4}+....+\frac{1}{2012}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{2013}\right)-2\left(\frac{1}{2}+\frac{1}{4}+.......+\frac{1}{2012}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{2013}\right)-\left(1+\frac{1}{2}+........+\frac{1}{1006}\right)\)
\(=\frac{1}{1007}+\frac{1}{1008}+......+\frac{1}{2013}\)
\(=P\)
\(\Leftrightarrow S-P=0\)
\(\Leftrightarrow\left(S-P\right)^{2013}=0\)
\(S=\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\frac{5}{6}+\frac{4}{5}-\frac{3}{4}+\frac{2}{3}-\frac{1}{2}\)
\(S=\left(\frac{1}{2}-\frac{1}{2}\right)-\left(\frac{2}{3}-\frac{2}{3}\right)+\left(\frac{3}{4}-\frac{3}{4}\right)-\left(\frac{4}{5}-\frac{4}{5}\right)+\frac{5}{6}\)
\(S=0-0+0-0+\frac{5}{6}\)
\(S=0+\frac{5}{6}\)
\(S=\frac{5}{6}\)
Cẩn thận nha mấy bn, bài này dễ sai dấu lém đó ng kq vẫn đúng
Ủng hộ mk nha ^_^
\(S=\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\frac{5}{6}+\frac{4}{5}-\frac{3}{4}+\frac{2}{3}-\frac{1}{2}\)
\(\Rightarrow S=\left(\frac{1}{2}-\frac{1}{2}\right)+\left(\frac{3}{4}-\frac{3}{4}\right)-\left(\frac{4}{5}-\frac{4}{5}\right)-\left(\frac{2}{3}-\frac{2}{3}\right)+\frac{5}{6}\)
\(\Rightarrow S=\frac{5}{6}\)
\(2-\frac{2}{3+\frac{1}{2}}=2-\frac{2}{\frac{6}{2}+\frac{1}{2}}=2-\frac{2}{\frac{7}{2}}=2-2:\frac{7}{2}=2-\frac{4}{7}=\frac{10}{7}\)
\(\frac{1-\frac{1}{\frac{3}{4}+1}}{3}=1-\frac{1}{\frac{7}{\frac{4}{3}}}=1-\frac{1}{7}:\frac{4}{3}=1-\frac{3}{28}=\frac{25}{28}\)