\(\frac{2}{3\cdot7}+\frac{2}{7\cdot11}+...+\frac{2}{71\cdot75}+\frac{2}{75\cdot79}\)

\(=\frac{2}{4}\left[\frac{4}{3\cdot7}+\frac{4}{7\cdot11}+...+\frac{4}{71\cdot75}+\frac{4}{75\cdot79}\right]\)

\(=\frac{2}{4}\left[\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{75}-\frac{1}{79}\right]\)

\(=\frac{1}{2}\left[\frac{1}{3}-\frac{1}{79}\right]=\frac{38}{237}\)

       \(\frac{2}{3\cdot7}+\frac{2}{7\cdot11}+...+\frac{2}{71\cdot75}+\frac{2}{75\cdot79}\)

\(=\frac{1}{2}\left(\frac{4}{3\cdot7}+\frac{4}{7\cdot11}+...+\frac{4}{71\cdot75}+\frac{4}{75\cdot79}\right)\)

\(=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{71}-\frac{1}{75}+\frac{1}{75}-\frac{1}{79}\right)\)

\(=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{79}\right)\)

\(=\frac{1}{2}\cdot\frac{76}{237}\)

\(=\frac{38}{237}\)

#)Giải :

Đặt \(A=\frac{2}{3.7}+\frac{2}{7.11}+\frac{2}{11.15}+...+\frac{2}{75.79}\)

\(\Rightarrow2A=\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{75.79}\)

\(\Rightarrow2A=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{75}-\frac{1}{79}\)

\(\Rightarrow2A=\frac{1}{3}-\frac{1}{79}\)

\(\Rightarrow2A=\frac{76}{237}\)

\(\Rightarrow A=\frac{38}{237}\)

Giải:

A=2/3.7+2/7.11+2/11.15+...+2/n.(n+4)

A=1/2.(4/3.7+4/7.11+4/11.15+...+4/n.(n+4)

A=1/2.(1/3-1/7+1/7-1/11+1/11-1/15+...+1/n-1/n+4)

A=1/2.(1/3-1/n+4)

A=1/6-1/2.(n+4)

⇒A<1/6

Chúc bạn học tốt!

Ta có : \(A=\dfrac{2}{3.7}+\dfrac{2}{7.11}+...+\dfrac{2}{n\left(n+4\right)}\)

\(\Rightarrow4A=\dfrac{8}{3.7}+\dfrac{8}{7.11}+...+\dfrac{8}{n\left(n+4\right)}\)

\(\Rightarrow4A=\dfrac{8}{3.7}+\dfrac{8}{7.11}+...+\dfrac{8}{n\left(n+4\right)}\)\(=\dfrac{2}{3}-\dfrac{2}{7}+\dfrac{2}{7}-\dfrac{2}{11}+...+\dfrac{2}{n}-\dfrac{2}{n+4}=\dfrac{2}{3}-\dfrac{2}{n+4}\)

\(\Rightarrow A=\dfrac{1}{6}-\dfrac{1}{2\left(n+4\right)}\)

- Xét hiệu \(A-\dfrac{1}{6}=-\dfrac{1}{2\left(n+4\right)}< 0\)

Vậy A < 1/6

Chắc là đề thiếu: \(y=\frac{1}{2}-\frac{1}{3\cdot7}-\frac{1}{7\cdot11}-\frac{1}{11\cdot15}-\frac{1}{15\cdot19}-\frac{1}{19\cdot23}-\frac{1}{23\cdot27}\)

\(y=\frac{1}{2}-\left(\frac{1}{3\cdot7}+\frac{1}{7\cdot11}+...+\frac{1}{23\cdot27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}\left(\frac{4}{3\cdot7}+\frac{4}{7\cdot11}+...+\frac{4}{23\cdot27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{23}-\frac{1}{27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}\left(\frac{1}{3}-\frac{1}{27}\right)=\frac{1}{2}-\frac{1}{4}\cdot\frac{8}{27}=\frac{23}{54}\)

Ta có:

\(\frac{2}{3.7}+\frac{2}{7.11}+\frac{2}{11.15}+.....+\frac{2}{2015.2019}\)

\(=2.\left(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+....+\frac{1}{2015.2019}\right)\)

\(=\frac{2}{4}.\left(\frac{2}{3.7}+\frac{2}{7.11}+\frac{2}{11.15}+....+\frac{2}{2015.2019}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+.....+\frac{1}{2015}-\frac{1}{2019}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{2019}\right)\)

\(=\frac{1}{2}.\frac{224}{673}=\frac{112}{673}\)

**** ^^

Mình cũng làm như bạn vậy

A=\(\frac{-2}{3.7}\)+\(\frac{-2}{7.11}\)+\(\frac{-2}{11.15}\)+....+\(\frac{-2}{97.101}\)

A=\(\frac{-1}{2}\).(\(\frac{4}{3.7}\)+\(\frac{4}{7.11}\)+\(\frac{4}{11.15}\)+.....+\(\frac{4}{97.101}\))

A=\(\frac{-1}{2}\)(\(\frac{1}{3}\)-\(\frac{1}{7}\)+\(\frac{1}{7}\)-\(\frac{1}{11}\)+\(\frac{1}{11}\)-\(\frac{1}{15}\)+....+\(\frac{1}{97}\)-\(\frac{1}{101}\))

A=\(\frac{-1}{2}\).(\(\frac{1}{3}\)-\(\frac{1}{101}\))

A=\(\frac{-1}{2}\).\(\frac{104}{303}\)=\(\frac{-52}{303}\)

Ta có : \(\frac{1}{2}-\frac{1}{3.7}-\frac{1}{7.11}-\frac{1}{11.15}-\frac{1}{15.19}-\frac{1}{19.23}-\frac{1}{23.27}\)

\(=\frac{1}{2}-\left(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+\frac{1}{15.19}+\frac{1}{19.23}+\frac{1}{23.27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}\left(\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+\frac{4}{15.19}+\frac{4}{19.23}+\frac{4}{23.27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+\frac{1}{15}-\frac{1}{19}+\frac{1}{19}-\frac{1}{23}+\frac{1}{23}-\frac{1}{27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{27}\right)=\frac{1}{2}-\frac{1}{4}.\frac{8}{27}=\frac{1}{2}-\frac{2}{27}=\frac{23}{54}\)

Trả lời:

\(\frac{1}{2}-\frac{1}{3.7}-\frac{1}{7.11}-\frac{1}{11.15}-\frac{1}{15.19}-\frac{1}{19.23}-\frac{1}{23.27}\)

\(=\frac{1}{2}-\left(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+\frac{1}{15.19}+\frac{1}{19.23}+\frac{1}{23.27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}.\left(\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+\frac{4}{15.19}+\frac{4}{19.23}+\frac{4}{23.27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+\frac{1}{15}-\frac{1}{19}+\frac{1}{19}-\frac{1}{23}+\frac{1}{23}-\frac{1}{27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{27}\right)\)

\(=\frac{1}{2}-\frac{1}{4}.\frac{8}{27}\)

\(=\frac{1}{2}-\frac{2}{27}\)

\(=\frac{23}{54}\)

Học tốt 

a.\(A=\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{107.111}\)

\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{107}-\frac{1}{111}\)

\(=\frac{1}{3}-\frac{1}{111}=\frac{37}{111}-\frac{1}{111}=\frac{36}{111}=\frac{12}{37}\)

Vậy A=\(\frac{12}{37}\)

b.\(B=\frac{2}{15}+\frac{2}{35}+\frac{2}{63}+...+\frac{2}{399}\)

\(=\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{19.21}\)

\(=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{19}-\frac{1}{21}\)

\(=\frac{1}{3}-\frac{1}{21}=\frac{7}{21}-\frac{1}{21}=\frac{6}{21}=\frac{2}{7}\)

Vậy \(B=\frac{2}{7}\)

c.\(C=\frac{1}{10}+\frac{1}{15}+...+\frac{1}{120}\)

\(\Rightarrow C.\frac{1}{2}=\left(\frac{1}{10}+\frac{1}{15}+...+\frac{1}{120}\right).\frac{1}{2}\)

\(=\frac{1}{20}+\frac{1}{30}+...+\frac{1}{240}\)

\(=\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{15.16}\)

\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{15}-\frac{1}{16}\)

\(=\frac{1}{4}-\frac{1}{16}=\frac{4}{16}-\frac{1}{16}=\frac{3}{16}\)

Vậy \(C=\frac{3}{16}\)

A = \(\frac{4}{3.7}+\frac{4}{7.9}+...+\frac{4}{107.111}\)

A = \(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{107}-\frac{1}{111}\)

A = \(\frac{1}{3}-\frac{1}{111}\)=\(\frac{12}{37}\)

2 câu sau tương tự. Mik ngại làm lắm -_-

\(A=\frac{4^2}{3.7}+\frac{4^2}{7.11}+\frac{4^2}{11.15}+...+\frac{4^2}{107.111}\)

\(A=\) \(4\left(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+...+\frac{1}{107.111}\right)\)

\(A=4\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{107}-\frac{1}{111}\right)\)

\(A=4\left(\frac{1}{3}-\frac{1}{111}\right)\)

\(A=4.\frac{12}{37}\)

\(A=\frac{48}{37}\)

lớp 6