\(\frac{1}{20\cdot23}+\frac{1}{23\cdot26}+...+\frac{1}{77\cdot80}\)

\(=\frac{1}{3}\left[\frac{3}{20\cdot23}+\frac{3}{23\cdot26}+...+\frac{3}{77\cdot80}\right]\)

\(=\frac{1}{3}\left[\frac{1}{20}-\frac{1}{23}+...+\frac{1}{77}-\frac{1}{80}\right]\)

\(=\frac{1}{3}\left[\frac{1}{20}-\frac{1}{80}\right]\)

\(=\frac{1}{3}\left[\frac{4}{80}-\frac{1}{80}\right]\)

\(=\frac{1}{3}\cdot\frac{3}{80}=\frac{1}{1}\cdot\frac{1}{80}=\frac{1}{80}\)

Mà \(\frac{1}{80}< \frac{1}{9}\)nên \(\frac{1}{20\cdot23}+\frac{1}{23\cdot26}+...+\frac{1}{77\cdot80}< \frac{1}{9}\)

Vậy : ...

\(\frac{1}{20.23}+\frac{1}{23.26}+...+\frac{1}{77.80}\)

\(=\frac{1}{3}.\left(\frac{3}{20.23}+\frac{3}{23.26}+...+\frac{3}{77.80}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{20}-\frac{1}{23}+\frac{1}{23}-\frac{1}{26}+...+\frac{1}{77}-\frac{1}{80}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{20}-\frac{1}{80}\right)\)

\(=\frac{1}{3}.\frac{3}{80}\)

\(=\frac{1}{80}< \frac{1}{9}\)

Đặt \(A=\frac{3^2}{20.23}+\frac{3^2}{23.26}+...+\frac{3^2}{77.80}\) ta có : 

\(A=3\left(\frac{3}{20.23}+\frac{3}{23.26}+...+\frac{3}{77.80}\right)\)

\(A=3\left(\frac{1}{20}-\frac{1}{23}+\frac{1}{23}-\frac{1}{26}+...+\frac{1}{77}-\frac{1}{80}\right)\)

\(A=3\left(\frac{1}{20}-\frac{1}{80}\right)\)

\(A=3.\frac{3}{80}\)

\(A=\frac{9}{80}< 1\) ( tử bé hơn mẫu ) 

Vậy \(A< 1\)

Chúc bạn học tốt ~ 

đặt A=9/20x23+9/23x26+...+9/77x80

<=>\(A=3\left(\frac{3}{20\cdot23}+\frac{3}{23\cdot26}+...+\frac{3}{77\cdot80}\right)\)

\(\Rightarrow A=3\left(\frac{1}{20}-\frac{1}{23}+\frac{1}{23}-\frac{1}{26}+...+\frac{1}{77}-\frac{1}{80}\right)\)

\(\Rightarrow A=3\left(\frac{1}{20}-\frac{1}{80}\right)\)

\(\Rightarrow A=3\cdot\frac{3}{80}\)

\(\Rightarrow A=\frac{9}{80}\)

đặt A=3²/20x23+3²/23x26+...+3²/77x80

\(A=3\left(\frac{3}{20\cdot23}+\frac{3}{23\cdot26}+...+\frac{3}{77\cdot80}\right)\)

\(=3\left(\frac{1}{20}-\frac{1}{23}+\frac{1}{23}-\frac{1}{26}+...+\frac{1}{77}-\frac{1}{80}\right)\)

\(=3\left(\frac{1}{20}-\frac{1}{80}\right)\)

\(=3\cdot\frac{3}{80}\)

\(=\frac{9}{80}\)

giúp với

da la dau

a) \(\frac{1}{1x3}+\frac{1}{3x5}+\frac{1}{5x7}+...+\frac{1}{2007x2009}\)

\(=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2007}-\frac{1}{2009}\right)\)

\(=\frac{1}{2}.\left(1-\frac{1}{2009}\right)=\frac{1}{2}\cdot\frac{2008}{2009}=\frac{1004}{2009}\)

....

các bài cn lại bn lm tương tự nha

b, \(\dfrac{1}{18}+\dfrac{1}{54}+\dfrac{1}{108}+...+\dfrac{1}{990}\)

3A = \(\dfrac{1}{6}+\dfrac{1}{18}+...+\dfrac{1}{330}\)

3A-A = \(\dfrac{1}{6}-\dfrac{1}{990}\)

2A = 82/495

A =82/495 : 2 

A=41/495

c, \(\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2008.2010}\)

A= \(\dfrac{4}{2}.\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2008.2010}\right)\)

A= \(\dfrac{4}{2}.\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2008}-\dfrac{1}{2010}\right)\)

A= \(\dfrac{4}{2}.\left(\dfrac{1}{2}-\dfrac{1}{1010}\right)\)

A= \(\dfrac{4}{2}.\dfrac{252}{505}\)

A= \(\dfrac{504}{505}\)

\(\frac{1}{11×14}+\frac{1}{14×17}+\frac{1}{17×20}+\frac{1}{20×23}+\frac{1}{23×26}\)

\(=\frac{1}{3}×\left(\frac{3}{11×14}+\frac{3}{14×17}+\frac{3}{17×20}+\frac{3}{20×23}+\frac{3}{23×26}\right)\)

\(=\frac{1}{3}×\left(\frac{1}{11}-\frac{1}{14}+\frac{1}{14}-\frac{1}{17}+...+\frac{1}{23}-\frac{1}{26}\right)\)

\(=\frac{1}{3}×\left(\frac{1}{11}-\frac{1}{26}\right)\)

\(=\frac{1}{3}×\frac{15}{286}\)

\(=\frac{5}{286}\)

\(\frac{1}{11\times14}+\frac{1}{14\times17}+\frac{1}{17\times20}+\frac{1}{20\times23}+\frac{1}{23\times26}\)

\(=\frac{1}{3}\times\left(\frac{1}{11\times14}+\frac{1}{14\times17}+\frac{1}{17\times20}+\frac{1}{20\times23}+\frac{1}{23\times26}\right)\)

\(=\frac{1}{3}\times\left(\frac{1}{11}-\frac{1}{14}+\frac{1}{14}-\frac{1}{17}+\frac{1}{17}-\frac{1}{20}+\frac{1}{20}-\frac{1}{23}+\frac{1}{23}-\frac{1}{26}\right)\)

\(=\frac{1}{3}\times\left(\frac{1}{11}-\frac{1}{26}\right)\)

  =    5/286

Đặt biểu thức trên là A. Ta có :

3A = \(\frac{3}{11x14}+\frac{3}{14x17}+\frac{3}{17x20}+\frac{3}{20x23}+\frac{3}{23x26}\)

3A = \(\frac{1}{11}-\frac{1}{14}+\frac{1}{14}-\frac{1}{17}+\frac{1}{17}-\frac{1}{20}+\frac{1}{23}-\frac{1}{26}\)

3A = \(\frac{1}{11}-\frac{1}{26}=\frac{15}{286}\)

A = \(\frac{15}{286}:3=\frac{5}{286}\)

                                  Vậy A = \(\frac{15}{286}\)

a) \(\frac{3}{4\times9}+\frac{3}{9\times14}+...+\frac{3}{54\times59}+\frac{3}{59\times64}\)

\(=\frac{3}{5}\times\left(\frac{5}{4\times9}+\frac{5}{9\times14}+...+\frac{5}{59\times64}\right)\)

\(=\frac{3}{5}\times\left(\frac{9-4}{4\times9}+\frac{14-9}{9\times14}+...+\frac{64-59}{59\times64}\right)\)

\(=\frac{3}{5}\times\left(\frac{1}{4}-\frac{1}{9}+\frac{1}{9}-\frac{1}{14}+...+\frac{1}{59}-\frac{1}{64}\right)\)

\(=\frac{3}{5}\times\left(\frac{1}{4}-\frac{1}{64}\right)\)

\(=\frac{9}{64}\)

b) \(\frac{2}{8\times11}+\frac{2}{11\times14}+...+\frac{2}{23\times26}+\frac{2}{26\times29}\)

\(=\frac{2}{3}\times\left(\frac{3}{8\times11}+\frac{3}{11\times14}+...+\frac{3}{26\times29}\right)\)

\(=\frac{2}{3}\times\left(\frac{11-8}{8\times11}+\frac{14-11}{11\times14}+...+\frac{29-26}{26\times29}\right)\)

\(=\frac{2}{3}\times\left(\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+...+\frac{1}{26}-\frac{1}{29}\right)\)

\(=\frac{2}{3}\times\left(\frac{1}{8}-\frac{1}{29}\right)\)

\(=\frac{7}{116}\)