5/7 : 9/11 + 9/7 : 9/11 ..... 3 : 9/11
Điền > , <, =.
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\(\frac{1}{3}-\frac{3}{5}+\frac{5}{7}-\frac{7}{9}+\frac{9}{11}-\frac{11}{13}+\frac{13}{15}+\frac{11}{13}-\frac{9}{11}+\frac{7}{9}-\frac{5}{7}+\frac{3}{5}\)
\(=\left(-\frac{3}{5}+\frac{3}{5}\right)+\left(\frac{5}{7}-\frac{5}{7}\right)+\left(-\frac{7}{9}+\frac{7}{9}\right)+\left(\frac{9}{11}-\frac{9}{11}\right)+\left(-\frac{11}{13}+\frac{11}{13}\right)+\left(\frac{1}{3}+\frac{13}{15}\right)\)
\(=\frac{1}{3}+\frac{13}{15}\)
\(=\frac{6}{5}\)
\(\frac{1}{3}-\frac{3}{5}+\frac{5}{7}-\frac{7}{9}+\frac{9}{11}-\frac{11}{13}-\frac{9}{11}+\frac{7}{9}-\frac{5}{7}+\frac{3}{5}-\frac{1}{3}\)
\(=\left(\frac{1}{3}-\frac{1}{3}\right)+\left(\frac{3}{5}-\frac{3}{5}\right)+\left(\frac{5}{7}-\frac{5}{7}\right)+\left(\frac{7}{9}-\frac{7}{9}\right)+\left(\frac{9}{11}-\frac{9}{11}\right)-\frac{11}{13}\)
\(=0+0+0+0+0-\frac{11}{13}=-\frac{11}{13}\)
\(\frac{1}{3}-\frac{3}{5}+\frac{5}{7}-\frac{7}{9}+\frac{9}{11}+\frac{7}{9}-\frac{5}{7}+\frac{3}{5}-\frac{1}{3}\)
\(=\left(\frac{1}{3}-\frac{1}{3}\right)+\left(\frac{3}{5}-\frac{3}{5}\right)+\left(\frac{5}{7}-\frac{5}{7}\right)+\left(\frac{7}{9}-\frac{7}{9}\right)-\frac{11}{13}\)
\(=0+0+0+0+0-\frac{11}{13}=-\frac{11}{13}\)
A= (1/3 - 1/3) + (-3/5 + 3/5) + (5/7 - 5/7) + (-7/9 + 7/9) + (9/11 - 9/11) - 11/13
A= -11/13
\(A=\dfrac{1}{3}-\dfrac{3}{5}+\dfrac{5}{7}-\dfrac{7}{9}+\dfrac{9}{11}-\dfrac{11}{13}-\dfrac{9}{11}+\dfrac{7}{9}-\dfrac{5}{7}+\dfrac{3}{5}-\dfrac{1}{3}\\ \Leftrightarrow A=\dfrac{1}{3}-\dfrac{1}{3}-\dfrac{3}{5}+\dfrac{3}{5}+\dfrac{5}{7}-\dfrac{5}{7}-\dfrac{7}{9}+\dfrac{7}{9}+\dfrac{9}{11}-\dfrac{9}{11}-\dfrac{11}{13}\\ \Leftrightarrow A=\left(\dfrac{1}{3}-\dfrac{1}{3}\right)-\left(\dfrac{3}{5}-\dfrac{3}{5}\right)+\left(\dfrac{5}{7}-\dfrac{5}{7}\right)-\left(\dfrac{7}{9}-\dfrac{7}{9}\right)+\left(\dfrac{9}{11}-\dfrac{9}{11}\right)-\dfrac{11}{13}\\ \Leftrightarrow A=0-0+0-0+0-\dfrac{11}{13}\\ \Leftrightarrow A=\dfrac{-11}{13}\)
5/7:9/11+9/7:9/11=5/7*11/9+9/7*11/9=(5/7+9/7)*11/9=14/7*11/9=2*11/9=22/9
3:9/11=3*11/9=33/9
Vì 22<33 nên 22/9<33/9 hay 5/7:9/11+9/7:9/11<3:9/11