\(\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+\dfrac{1}{7\cdot9}+...+\dfrac{1}{x\left(x+2\right)}=\dfrac{11}{75}\left(x\ne0;x\ne-2\right)\)
\(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}+...+\dfrac{2}{x\left(x+2\right)}=\dfrac{22}{75}\)
\(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{x}-\dfrac{1}{x+2}=\dfrac{22}{75}\)
\(\dfrac{1}{3}-\dfrac{1}{x+2}=\dfrac{22}{75}\)
\(\dfrac{1}{x+2}=\dfrac{1}{3}-\dfrac{22}{75}\)
\(\dfrac{1}{x+2}=\dfrac{1}{25}\)
suy ra
\(x+2=25\\ x=23\left(tm\right)\)
=>2/3*5+2/5*7+...+2/x(x+2)=22/75
=>1/3-1/5+1/5-1/7+...+1/x-1/x+2=22/75
=>1/3-1/x+2=22/75
=>1/x+2=25/75-22/75=3/75=1/25
=>x=23
