Đáp án:
$7)\dfrac{2010}{2011}\\
8)
\dfrac{670}{2013}\\
9)
\dfrac{106}{319}\\
14)
\dfrac{91}{5}\\$
Giải thích các bước giải:
$7)\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2009.2010}+\dfrac{1}{2010.2011}\\
=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2009}-\dfrac{1}{2010}+\dfrac{1}{2010}-\dfrac{1}{2011}\\
=1-\dfrac{1}{2011}\\
=\dfrac{2011}{2011}-\dfrac{1}{2011}\\
=\dfrac{2010}{2011}\\
8)
\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+...+\dfrac{2}{2009.2011}.\dfrac{2}{2011.2013}\\
=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{2009}-\dfrac{1}{2011}+\dfrac{1}{2011}-\dfrac{1}{2013}\\
=\dfrac{1}{3}-\dfrac{1}{2013}\\
=\dfrac{671}{2013}-\dfrac{1}{2013}\\
=\dfrac{670}{2013}\\
9)
\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+...+\dfrac{1}{313.316}+\dfrac{1}{316.319}\\
=\dfrac{1}{3}.\left (\dfrac{3}{1.4}+\dfrac{3}{4.7}+\dfrac{3}{7.10}+...+\dfrac{3}{313.316}+\dfrac{3}{316.319} \right )\\
=\dfrac{1}{3}.\left ( \dfrac{1}{1}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{313}-\dfrac{1}{316} +\dfrac{1}{316}-\dfrac{1}{319}\right )\\
=\dfrac{1}{3}.\left ( 1-\dfrac{1}{319} \right )\\
=\dfrac{1}{3}.\left ( \dfrac{319}{319}-\dfrac{1}{319} \right )\\
=\dfrac{1}{3}.\dfrac{318}{319}\\
=\dfrac{106}{319}\\
14)
5\dfrac{3}{5}+1,75+6\dfrac{1}{8}+4\dfrac{1}{4}+3,875-3,4\\
=\dfrac{28}{5}+\dfrac{175}{100}+\dfrac{49}{8}+\dfrac{17}{4}+0,475\\
=\dfrac{28}{5}+\dfrac{7}{4}+\dfrac{49}{8}+\dfrac{17}{4}+\dfrac{19}{40}\\
=\dfrac{28}{5}+\left (\dfrac{7}{4} +\dfrac{17}{4} \right )+\dfrac{49}{8}+\dfrac{19}{40}\\
=\dfrac{28}{5}+\dfrac{24}{4}+\dfrac{49}{8}+\dfrac{19}{40}\\
=\dfrac{28}{5}+6+\dfrac{49}{8}+\dfrac{19}{40}\\
=\dfrac{28.8}{40}+\dfrac{6.40}{40}+\dfrac{49.5}{40}+\dfrac{19}{40}\\
=\dfrac{728}{40}\\
=\dfrac{91}{5}\\$
