x + 5 is the mean proportional between x + 2 and x + 9.

So, (x + 2), (x + 5) and (x + 9) are in continued proportion.

\[\begin{array}{*{35}{l}}

\left( x\text{ }+\text{ }2 \right):\text{ }\left( x\text{ }+\text{ }5 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }5 \right):\text{ }\left( x\text{ }+\text{ }9 \right)  \\

\left( x\text{ }+\text{ }2 \right)/\text{ }\left( x\text{ }+\text{ }5 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }5 \right)/\text{ }\left( x\text{ }+\text{ }9 \right)  \\

{{\left( x\text{ }+\text{ }5 \right)}^{2}}~=\text{ }\left( x\text{ }+\text{ }2 \right)\left( x\text{ }+\text{ }9 \right)  \\

{{x}^{2}}~+\text{ }25\text{ }+\text{ }10x\text{ }=\text{ }{{x}^{2}}~+\text{ }2x\text{ }+\text{ }9x\text{ }+\text{ }18  \\

25\text{ }\text{ }18\text{ }=\text{ }11x\text{ }-\text{ }10x  \\

x\text{ }=\text{ }7  \\

\end{array}\]