nếu lm sai thì thôi na
đặt vế trái lak A
4A=5(5/3.7+5/7.11+.....+5/(4n-1)(4n+3)
4A=(1/3-1/4n+3)
4A=4n+3-3/3(4n+3)
4A=4/12n+9
A=5/4n+3
\(\frac{5}{3.7}+\frac{5}{7.11}+...+\frac{5}{\left(4n-1\right).\left(4n+3\right)}=\frac{5}{4n+3}\)
\(\Leftrightarrow\frac{5}{4}\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{4n-1}-\frac{1}{4n+3}\right)=\frac{5}{4n+3}\)
\(\Leftrightarrow\frac{1}{3}-\frac{1}{4n+3}=\frac{5}{4n+3}\cdot\frac{4}{5}\)
\(\Leftrightarrow\frac{1}{3}-\frac{1}{4n+3}=\frac{5\cdot4}{5\left(4n+3\right)}\)
\(\Leftrightarrow\frac{1}{3}-\frac{1}{4n+3}=\frac{4}{4n+3}\)
\(\Leftrightarrow\frac{1}{4n+3}=\frac{1}{3}-\frac{4}{4n+3}\)
\(\Leftrightarrow\frac{1}{4n+3}=\frac{4n+3-12}{3\left(4n+3\right)}\)
\(\Leftrightarrow\frac{1}{4n+3}=\frac{4n-9}{12n+9}\)
\(\Leftrightarrow\frac{3}{12n+9}=\frac{4n-9}{12n+9}\)
\(\Leftrightarrow4n-9=3\)
\(\Leftrightarrow4n=12\Leftrightarrow n=3\)
Vậy n = 3
