\(\dfrac{x}{2}+\dfrac{x}{6}+\dfrac{x}{12}+...+\dfrac{x}{90}=\dfrac{p^3+10}{p^2+2}\\ x\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{90}\right)=\dfrac{p^3+10}{p^3+2}\\ x\cdot\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{9\cdot10}\right)=\dfrac{p^3+10}{p^3+2}\\ x\cdot\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{9}-\dfrac{1}{10}\right)=\dfrac{p^3+10}{p^3+2}\\ x\cdot\left(1-\dfrac{1}{10}\right)=\dfrac{p^3+10}{p^3+2}\\ x\cdot\dfrac{9}{10}=\dfrac{p^3+10}{p^3+2}\\ x=\dfrac{p^3+10}{p^3+2}:\dfrac{9}{10}\\x=\dfrac{p^3+10}{p^3+2}\cdot\dfrac{10}{9}=\dfrac{10p^3+100}{9p^3+18}\)
\(\begin{align*} \frac{x}{2}+\frac{x}{6}+\frac{x}{12}+...+\frac{x}{90}&=\frac{p^{3}+10}{p^{2}+2} \\ x(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{90})&=\frac{p^{3}+10}{p^{3}+2} \\ x\cdot(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{9\cdot10})&=\frac{p^{3}+10}{p^{3}+2} \\ x\cdot(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{9}-\frac{1}{10})&=\frac{p^{3}+10}{p^{3}+2} \\ x\cdot(1-\frac{1}{10})&=\frac{p^{3}+10}{p^{3}+2} \\ x\cdot\frac{9}{10}&=\frac{p^{3}+10}{p^{3}+2} \\ p^{3} + 10 &= p^{3} + 2 \\ 9 &= 10 \\ x &= \frac{p^{3}+10}{p^{3}+2}\cdot\frac{10}{9} \\ x &= \frac{10p^{3}+100}{9p^{3}+18} \end{align*}\)
