Question

1/3*5 + 1/5*7 + 1/7*9 +...+1/ x(x+2)=11/75

Answer 1

\(\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+\dfrac{1}{7\cdot9}+...+\dfrac{1}{x\left(x+2\right)}=\dfrac{11}{75}\left(x\ne0;x\ne-2\right)\)

\(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}+...+\dfrac{2}{x\left(x+2\right)}=\dfrac{22}{75}\)

\(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{x}-\dfrac{1}{x+2}=\dfrac{22}{75}\)

\(\dfrac{1}{3}-\dfrac{1}{x+2}=\dfrac{22}{75}\)

\(\dfrac{1}{x+2}=\dfrac{1}{3}-\dfrac{22}{75}\)

\(\dfrac{1}{x+2}=\dfrac{1}{25}\)

suy ra

\(x+2=25\\ x=23\left(tm\right)\)

 

 

 

Answer 2

=>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