Đặt \(S=\)\(\dfrac{1}{3^2}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{20}}\)
\(\Rightarrow9S=1+\dfrac{1}{3^2}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{18}}\)
\(\Rightarrow9S-S=\left(1+\dfrac{1}{3^2}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{18}}\right)-\left(\dfrac{1}{3^2}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{20}}\right)\)
\(\Rightarrow8S=1-\dfrac{1}{3^{20}}\)
\(\Rightarrow S=\dfrac{1-\dfrac{1}{3^{20}}}{8}\)

\(A=\dfrac{9}{1.2}+\dfrac{9}{2.3}+\dfrac{9}{3.4}+...+\dfrac{9}{98.99}+\dfrac{9}{99.100}\)
\(=9\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{98.99}+\dfrac{1}{99.100}\right)\)
\(=9\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{98}-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(=9\left(1-\dfrac{1}{100}\right)\)
\(=9.\dfrac{99}{100}\)
\(=\dfrac{891}{100}\)

\(A=\dfrac{9}{1.2}+\dfrac{9}{2.3}+\dfrac{9}{3.4}+...+\dfrac{9}{98.99}+\dfrac{9}{99.100}\)

\(=9.\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{98.99}+\dfrac{1}{99.100}\right)\)

\(=9.\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\)

\(=9.\left(1-\dfrac{1}{100}\right)\)

\(=9.\dfrac{99}{100}\)

\(=\dfrac{891}{100}\).

\(A=\left(1-\dfrac{1}{2}\right).\left(1-\dfrac{1}{3}\right).\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{30}\right)\)

\(=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}...\dfrac{29}{30}\)

\(=\dfrac{1.2.3...29}{2.3.4...30}\)

\(=\dfrac{1}{30}\)

\(B=1\dfrac{1}{3}.1\dfrac{1}{8}.1\dfrac{1}{15}.1\dfrac{1}{24}...1\dfrac{1}{168}\)

\(=\dfrac{4}{3}.\dfrac{9}{8}.\dfrac{16}{15}.\dfrac{25}{24}...\dfrac{169}{168}\)

\(=\dfrac{4.9.16.25...169}{3.8.15.24...168}\)

\(=\dfrac{2.3.4...13}{1.2.3...12}.\dfrac{2.3.4...13}{3.4.5...14}\)

\(=13.\dfrac{1}{7}\)

\(=\dfrac{13}{7}\).

A = \(\dfrac{1}{2}x\dfrac{2}{3}x\dfrac{3}{4}x...x\dfrac{29}{30}=1x1x1x...x\dfrac{1}{30}=\dfrac{1}{30}\)

B = \(\dfrac{4}{3}x\dfrac{9}{8}x\dfrac{16}{15}x\dfrac{25}{24}x...x\dfrac{169}{168}=1x1x1x1x...x\dfrac{13}{7}=\dfrac{13}{7}\)

Câu B em chưa rõ cách làm nhanh cho lắm. Nếu ko cần tính nhanh thì chị có thể giải bình thường ra giấy ha.

\(\dfrac{1}{x-1}+\dfrac{-2}{3}\left(\dfrac{3}{4}-\dfrac{6}{5}\right)=\dfrac{5}{2-2x}\)

\(\dfrac{1}{x-1}+\dfrac{-2}{3}\times\dfrac{-9}{20}=\dfrac{5}{2-2x}\)

\(\dfrac{1}{x-1}+\dfrac{3}{10}=\dfrac{5}{2-2x}\)

\(\dfrac{10+3x-3}{10x-10}=\dfrac{5}{2-2x}\)

\(\dfrac{7+3x}{10x-10}=\dfrac{5}{2-2x}\)

\(\left(7+3x\right)\left(2-2x\right)=5\left(10x-10\right)\)

\(7\left(2-2x\right)+3x\left(2-2x\right)=50x-50\)

\(14-14x+6x-6x^2=50x-50\)

\(14-8x-6x^2=50x-50\)

\(14-8x-6x^2+50=50x\)

\(36+8x-6x^2=50x\)

\(36=50x+6x^2-8x\)

\(36=42x+6x^2\)

\(\dfrac{36}{6}=\dfrac{42x+6x^2}{6}\)

\(6=7x+x^2\)

\(6=x\left(7+x\right)\)

còn lại mình chịu

\(\dfrac{1}{x-1}-\dfrac{2}{3}.\left(\dfrac{3}{4}-\dfrac{6}{5}\right)=\dfrac{5}{2-2x}\) (*)

ĐKXĐ: \(x\ne1\)

(*) \(\Leftrightarrow\dfrac{1}{x-1}+\dfrac{3}{10}=-\dfrac{5}{2x-2}\)

\(\Leftrightarrow\dfrac{1}{x-1}+\dfrac{5}{2x-2}=-\dfrac{3}{10}\)

\(\Leftrightarrow\dfrac{1.2}{\left(x-1\right).2}+\dfrac{5}{2x-2}=-\dfrac{3}{10}\)

\(\Leftrightarrow\dfrac{7}{2x-2}=-\dfrac{3}{10}\)

\(\Leftrightarrow2x-2=-\dfrac{70}{3}\)

\(\Leftrightarrow2x=-\dfrac{64}{3}\)

\(\Leftrightarrow x=-\dfrac{32}{3}\)

Vậy \(x=-\dfrac{32}{3}\).

\(\dfrac{x}{2}.\left(\dfrac{3x-13}{5}\right)=\dfrac{21x}{10}\)

\(\left(\dfrac{3x-13}{5}\right)=\dfrac{21x}{10}.\dfrac{2}{x}\)

\(\dfrac{3x-13}{5}=\dfrac{21}{5}\)

3x - 13 = 21 

3x = 34

x = 34/3

\(\dfrac{9}{2}-(\dfrac{-1}{3}x-\dfrac{17}{12})=(\dfrac{-2}{9})^2\)

\(\dfrac{9}{2}-(\dfrac{-1}{3}x-\dfrac{17}{12})=\dfrac{4}{81}\)

\(\dfrac{-1}{3}x-\dfrac{17}{12}=\dfrac{721}{162}\)

\(\dfrac{-1}{3}x=\dfrac{1901}{324}\)

\(x=\dfrac{-1901}{108}\)

\(\dfrac{9}{2}-\left(\dfrac{-1}{3}x-\dfrac{17}{12}\right)=\left(\dfrac{-2}{9}\right)^2\)

\(\dfrac{9}{2}-\left(\dfrac{-1}{3}x-\dfrac{17}{12}\right)=\dfrac{2}{9}\)

\(\dfrac{-1}{3}x-\dfrac{17}{12}=\dfrac{9}{2}-\dfrac{2}{9}=\dfrac{77}{18}\)

\(x=\dfrac{77}{18}:\dfrac{-1}{3}=-13\) 

`5/[x-2]=10/16`

`10(x-2)=16.5=80`

`x-2=80:10=8`

`x=8+2=10`

\(\dfrac{5}{x-2}=\dfrac{10}{16}\)

\(\Rightarrow5.16=\left(x-2\right).10\)

\(\Rightarrow\left(x-2\right).10=80\)

\(\Rightarrow x-2=80:10\)

\(\Rightarrow x-2=8\)

\(\Rightarrow x=8+2\)

\(\Rightarrow x=10\)

`2/4 =x/8`

`=> 2.8=4.x`

`=> 4x= 16`

`=>x=16:4`

`=>x=4`

2 phần 4 = x phần 8 bạn nhé

`(2x+4) xx 2=7/3`

`=>2x+4 = 7/3 : 2`

`=>2x+4 = 7/3 xx 1/2`

`=>2x+4 = 7/6`

`=>2x= 7/6-4`

`=>2x= 7/6- 24/6`

=>2x=-17/6`

`=>x= -17/6 : 2`

`=>x=-17/6 xx 1/2`

`=> x=-17/12`

\(\dfrac{2}{3}\) x = \(\dfrac{4}{7}\)                                                        \(\dfrac{-6}{5}\) x = \(\dfrac{1}{2}\)

    x  = \(\dfrac{4}{7}\) : \(\dfrac{2}{3}\)                                                   x  = \(\dfrac{1}{2}\) : ( -\(\dfrac{6}{5}\))

  x     =    \(\dfrac{6}{7}\)                                                    x   =  \(-\dfrac{5}{12}\)       

c, \(\dfrac{2}{3}\)  x = - \(\dfrac{1}{4}\)                                     \(\dfrac{3}{4}\) x = \(\dfrac{2}{-7}\)

        x  = - \(\dfrac{1}{4}\) : \(\dfrac{2}{3}\)                                  x   = \(\dfrac{2}{-7}\) : \(\dfrac{3}{4}\)

       x  = \(\dfrac{-3}{8}\)                                          x   = - \(\dfrac{8}{21}\)

\(a,\dfrac{2}{3}.x=\dfrac{4}{7}\)

\(x=\dfrac{4}{7}:\dfrac{2}{3}\)

\(x=\dfrac{4}{7}.\dfrac{3}{2}\)

\(x=\dfrac{12}{14}=\dfrac{6}{7}\)

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\(b,\dfrac{-6}{5}.x=\dfrac{1}{2}\)

\(x=\dfrac{1}{2}:\dfrac{-6}{5}\)

\(x=\dfrac{1}{2}.\dfrac{-5}{6}\)

\(x=\dfrac{-5}{12}\)

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\(c,\dfrac{2}{3}.x=\dfrac{-1}{4}\)

\(x=\dfrac{-1}{4}:\dfrac{2}{3}\)

\(x=\dfrac{-1}{4}.\dfrac{3}{2}\)

\(x=\dfrac{-3}{8}\)

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\(d,\dfrac{3}{4}.x=\dfrac{2}{-7}\)

\(x=\dfrac{2}{-7}:\dfrac{3}{4}\)

\(x=\dfrac{2}{-7}.\dfrac{4}{3}\)

\(x=\dfrac{-8}{21}\)