b: \(\dfrac{7}{x}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{41\cdot45}=\dfrac{29}{45}\)
=>\(\dfrac{7}{x}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{41}-\dfrac{1}{45}=\dfrac{29}{45}\)
=>\(\dfrac{7}{x}=\dfrac{29}{45}+\dfrac{1}{45}-\dfrac{1}{5}=\dfrac{2}{3}-\dfrac{1}{5}=\dfrac{7}{15}\)
=>x=15
c: \(\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+...+\dfrac{1}{\left(2x+1\right)\left(2x+3\right)}=\dfrac{15}{93}\)
=>\(\dfrac{1}{2}\left(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{\left(2x+1\right)\left(2x+3\right)}\right)=\dfrac{5}{31}\)
=>\(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{2x+1}-\dfrac{1}{2x+3}=\dfrac{5}{31}\cdot2=\dfrac{10}{31}\)
=>\(\dfrac{1}{3}-\dfrac{1}{2x+3}=\dfrac{10}{31}\)
=>\(\dfrac{1}{2x+3}=\dfrac{1}{3}-\dfrac{10}{31}=\dfrac{31-30}{93}=\dfrac{1}{93}\)
=>2x+3=93
=>2x=90
=>x=90:2=45
