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\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{9}-\dfrac{1}{10}\)

\(=1-\dfrac{1}{10}=\dfrac{9}{10}\)

\(\dfrac{2}{1\times3}+\dfrac{2}{3\times5}+\dfrac{2}{5\times7}+\dfrac{2}{7\times9}+\dfrac{2}{9\times11}\)

\(=2\times\left(\dfrac{1}{1\times3}+\dfrac{1}{3\times5}+\dfrac{1}{5\times7}+\dfrac{1}{7\times9}+\dfrac{1}{9\times11}\right)\)

\(=2\times\dfrac{1}{2}\times\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}\right)\)

\(=1-\dfrac{1}{11}\)

\(=\dfrac{11}{11}-\dfrac{1}{11}\)

\(=\dfrac{10}{11}\)

\(\dfrac{2}{1\times3}+\dfrac{2}{3\times5}+\dfrac{2}{5\times7}+\dfrac{2}{7\times9}+\dfrac{2}{9\times11}\\ =1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}\\ =1-\dfrac{1}{11}\\ =\dfrac{10}{11}\)

\(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{9\cdot11}\)

\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{9}-\dfrac{2}{11}\)

\(=1-\dfrac{2}{11}\)

\(=\dfrac{9}{11}\)

a)P=\(\dfrac{-2}{7}+\dfrac{14}{29}+\dfrac{12}{33}+\dfrac{15}{29}+\dfrac{21}{33}+\dfrac{2}{7}\)

      =\(\left(\dfrac{-2}{7}+\dfrac{2}{7}\right)+\left(\dfrac{14}{29}+\dfrac{15}{29}\right)+\left(\dfrac{12}{33}+\dfrac{21}{33}\right)\)

      =0+1+1=2

b)\(\dfrac{2}{7}.\dfrac{5}{19}+\dfrac{2}{7}.\dfrac{14}{19}+\dfrac{21}{19}-\dfrac{2}{7}.\dfrac{1}{5}\)

=\(\dfrac{2}{7}.\left(\dfrac{5}{19}+\dfrac{14}{19}-\dfrac{1}{5}\right)+\dfrac{21}{19}\)

=\(\dfrac{2}{7}.\dfrac{4}{5}+\dfrac{21}{19}=\dfrac{887}{665}\)

D

\(=1-\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{19}-\dfrac{1}{21}\right)\)

\(=1-\dfrac{6}{21}=\dfrac{15}{21}=\dfrac{5}{7}\)

a) \(\dfrac{3}{5}\times\dfrac{17}{21}+\dfrac{2}{5}\times\dfrac{17}{21}\)

\(=\dfrac{17}{21}\times\left(\dfrac{3}{5}+\dfrac{2}{5}\right)\)

\(=\dfrac{17}{21}\times1\)

\(=\dfrac{17}{21}\)

b) \(\dfrac{11}{19}\times\dfrac{2}{7}+\dfrac{5}{7}\times\dfrac{11}{19}\)

\(=\dfrac{11}{19}\times\left(\dfrac{2}{7}+\dfrac{5}{7}\right)\)

\(=\dfrac{11}{19}\times1\)

\(=\dfrac{11}{19}\)

a: =17/21(3/5+2/5)

=17/21*5/5

=17/21

b: =11/19(2/7+5/7)

=11/19*1

=11/19

`2/[3.5]+2/[5.7]+2/[7.9]+....+2/[19.21]`

`=1/3-1/5+1/5-1/7+1/7-1/9+....+1/19-1/21`

`=1/3-1/21`

`=6/21`

f)\(-\dfrac{2}{17}+\dfrac{15}{23}+-\dfrac{15}{17}+\dfrac{4}{19}+\dfrac{8}{23}\)
\(=\left(-\dfrac{2}{17}+-\dfrac{15}{17}\right)+\left(\dfrac{15}{23}+\dfrac{8}{23}\right)+\dfrac{4}{19}\)
\(=-1+1+\dfrac{4}{19}\)
\(=0+\dfrac{4}{19}=\dfrac{4}{19}\)
g)\(-\dfrac{1}{2}+\dfrac{3}{21}+-\dfrac{2}{6}+-\dfrac{5}{30}\)
\(=-\dfrac{1}{2}+\dfrac{1}{7}+-\dfrac{1}{3}+-\dfrac{1}{6}\)
\(=\left(-\dfrac{1}{2}+-\dfrac{1}{3}+-\dfrac{1}{6}\right)+\dfrac{1}{7}\)
\(=-\dfrac{3+2+1}{6}+\dfrac{1}{7}\)
\(=\dfrac{1}{7}-1\)
\(=\dfrac{1}{7}-\dfrac{7}{7}=-\dfrac{6}{7}\)

`f)(-2)/17 + 15/23 + (-15)/17 + 4/19 + 8/23`

`= (-2/17+ -15/17)+(15/23+8/23)+4/19`

`= -1+1+4/19`

`= 0 +4/19`

`= 0`

`g)(-1)/2 + 3/21 + (-2)/6 + (-5)/30`

`= (-1)/2 + 1/7 + (-1)/3 + (-1)/6`

`= (-21)/42 + 6/42 + (-14)/42 + (-7)/42`

`=(-36)/42`

`=(-6)/7`

\(f,-\dfrac{2}{17}+\dfrac{15}{23}-\dfrac{15}{17}+\dfrac{4}{19}+\dfrac{8}{23}\)

\(=-\dfrac{2}{17}-\dfrac{15}{17}+\dfrac{15}{23}+\dfrac{8}{23}+\dfrac{4}{19}\)

\(=-1+1+\dfrac{4}{19}\)

\(=\dfrac{4}{19}\)

\(g,-\dfrac{1}{2}+\dfrac{3}{21}-\dfrac{2}{6}-\dfrac{5}{30}\)

\(=-\dfrac{5}{14}-\dfrac{1}{2}\)

\(=-\dfrac{5}{14}-\dfrac{7}{14}\)

\(=-\dfrac{6}{7}\)

a) \(\left(1-\dfrac{1}{3}\right)\times\left(1-\dfrac{2}{5}\right)\times\left(1-\dfrac{2}{7}\right)\times\left(1-\dfrac{2}{9}\right)\)

\(=\left(\dfrac{3}{3}-\dfrac{1}{3}\right)\times\left(\dfrac{5}{5}-\dfrac{2}{5}\right)\times\left(\dfrac{7}{7}-\dfrac{2}{7}\right)\times\left(\dfrac{9}{9}-\dfrac{2}{9}\right)\)

\(=\dfrac{2}{3}\times\dfrac{3}{5}\times\dfrac{5}{7}\times\dfrac{7}{9}\)

\(=\dfrac{2\times3\times5\times7}{3\times5\times7\times9}\)

\(=\dfrac{2}{9}\)

b) \(\dfrac{1}{1\times3}+\dfrac{1}{3\times5}+\dfrac{1}{5\times7}+\dfrac{1}{7\times9}\)

\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}\)

\(=1-\dfrac{1}{9}\)

\(=\dfrac{9}{9}-\dfrac{1}{9}\)

\(=\dfrac{8}{9}\)