\(=\frac{3}{3.13}+\frac{3}{13.23}+...+\frac{3}{1993.2003}\)

\(=\frac{1}{10}.\left(1-\frac{3}{13}+\frac{3}{13}-\frac{3}{23}+...+\frac{3}{1993}-\frac{3}{2003}\right)\)

\(=\frac{1}{10}.\left(1-\frac{3}{2003}\right)\)

\(=\frac{1}{10}.\frac{2000}{2003}\)

\(=\frac{200}{2003}\)

Đặt \(A=\frac{1}{13}+\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{1993.2003}\)

\(\Rightarrow A=\frac{3}{3.13}+\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{1993.2003}\)

\(\Rightarrow A=3\left(\frac{1}{3.13}+\frac{1}{13.23}+\frac{1}{23.33}+...+\frac{1}{1993.2003}\right)\)

\(\Rightarrow A=\frac{3}{10}\left(\frac{10}{3.13}+\frac{10}{13.23}+\frac{10}{23.33}+...+\frac{10}{1993.2003}\right)\)

\(\Rightarrow A=\frac{3}{10}\left(\frac{1}{3}-\frac{1}{13}+\frac{1}{13}-\frac{1}{23}+\frac{1}{23}-\frac{1}{33}+...+\frac{1}{1993}-\frac{1}{2003}\right)\)

\(\Rightarrow A=\frac{3}{10}\left(\frac{1}{3}-\frac{1}{2003}\right)\)

\(\Rightarrow A=\frac{3}{10}.\left(\frac{2003}{6009}-\frac{3}{6009}\right)\)

\(\Rightarrow A=\frac{3}{10}.\frac{2000}{6009}\)

\(\Rightarrow A=\frac{200}{2003}\)

\(\frac{1}{13}+\frac{3}{13\cdot23}+\frac{3}{23\cdot33}+...+\frac{3}{1993\cdot2003}\)

\(=\frac{1}{13}+\left[\frac{3}{13\cdot23}+\frac{3}{23\cdot33}+...+\frac{3}{1993\cdot2003}\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\left[\frac{1}{13\cdot23}+\frac{1}{23\cdot33}+...+\frac{1}{1993\cdot2003}\right]\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\left[\frac{1}{13}-\frac{1}{23}+\frac{1}{23}-\frac{1}{33}+...+\frac{1}{1993}-\frac{1}{2003}\right]\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\left[\frac{1}{13}-\frac{1}{2003}\right]\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\cdot\frac{1990}{26039}\right]\)

\(=\frac{1}{13}+\frac{597}{26039}\)

\(=\frac{200}{2003}\)

Đặt A= 1/13 + 3/13.23 + 3/ 23.33 + ... + 3/1993.2003 

A- 1/13 = 3/13.23 + 3/ 23.33 + ... + 3/1993.2003 

10/3 ( A-1/3) =  10/3. (3/13.23 + 3/ 23.33 + ... + 3/1993.2003) 

10/3A - 10/9 = 10/13.23 + 10/ 23.33 + ... + 10/1993.2003 

10/3A - 10/9  = 1/13 - 1/23 + 1/23 - 1/33 +...+ 1/1993- 1/2003

10/3A = 1/13 - 1/2003 + 10/9

10/3 A= ? 

đến đây bn tự làm nha

10/3A - 10/9 = 1/13 

Ta loại \(\frac{1}{13}\)ra khỏi biểu thức để dễ tính hơn, sau đó cộng vào.

Gọi A là biểu thức \(\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{1993.2003}\)

Gấp A lên \(\frac{10}{3}\)lần, ta có :

\(A\times\frac{10}{3}=\frac{10}{3}\times\left(\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{1993.2003}\right)\)

\(A\times\frac{10}{3}=\frac{10}{13.23}+\frac{10}{23.33}+...+\frac{10}{1993.2003}\)

\(A\times\frac{10}{3}=\frac{1}{13}-\frac{1}{23}+\frac{1}{23}-\frac{1}{33}+...+\frac{1}{1993}-\frac{1}{2003}\)

\(A\times\frac{10}{3}=\frac{1}{13}-\frac{1}{2003}\)

\(A\times\frac{10}{3}=\frac{1990}{26039}\)

\(A=\frac{1990}{26039}\div\frac{10}{3}\)

\(A=\frac{597}{26039}\)

Biểu thức trên = \(\frac{597}{26039}+\frac{1}{13}=\frac{200}{2003}\)

a)\(\frac{2}{3}+\frac{3}{4}+\frac{5}{6}\)

\(=\frac{8+9+10}{12}\)

\(=\frac{27}{12}=\frac{9}{4}\)

b)\(\frac{15}{8}-\frac{7}{12}+\frac{5}{6}\)

\(=\frac{45-14+20}{24}\)

\(=\frac{51}{24}=\frac{17}{8}\)

2)

a)\(\frac{2}{5}+\frac{7}{13}+\frac{3}{5}+\frac{1}{7}\)

\(=\frac{2}{5}+\frac{3}{5}+\frac{7}{13}+\frac{1}{7}\)

\(=1+\frac{7}{13}+\frac{1}{7}\)

\(=\frac{20}{13}+\frac{1}{7}\)

\(=\frac{153}{91}\)

Tí tớ trả lời tiếp

lộn bạn đăng từng câu thôi

Co quy luat nay ne em: 1+2=3=2.3:2; 1+2+3=6=3.4:2;...;1+2+3+...+2012=2012.2013:2

Suy ra ta co:

Mau so cua D=1 + 1/(2.3:2)  +  1/(3.4:2)   +   1/(4.5:2)   +   ....   +   1/(2012.2013:2)

                    =1  +  2/2.3  +  2/3.4   +   2/4.5   +  ....  +   2/2012.2013

                    = 2.[1/2  +  1/2.3  +  1/3.4  +  1/4.5  +  .... +  1/2012.2013]

                    =2.[1/1.2   +  1/2.3   +   1/3.4   +  1/4.5   +  .....   +  1/2012.2013]

                    =2.[1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 +....+1/2012 - 1/2013

                    =2[1 - 1/2013]

                    =2.2012/2013

Vay D= 2.2012 / (2.2012:2013)=2013

\(\frac{3}{29}-\frac{1}{5}.\frac{29}{3}=\frac{3}{29}-\frac{29}{15}=\frac{-362}{145}\)

3/29 - 1/5 . 29/3

= 3/29 . 29/3 - 1/5

= 1 - 1/5

= 5/5 - 1/5

= 4/5

3/29 - 1/5 . 29/3

= 3/29 . 29/3 - 1/5

= 1 - 1/5

= 4/5

#hoc_tot#