10. Given an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference $d$ not equal to 0, and a geometric sequence $\left\{b_{n}\right\}$ with a common ratio $q$ that is a positive rational number less than 1. If $a_{1}=d, b_{1}=d^{2}$, and $\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{b_{1}+b_{2}+b_{3}}$ is a positive integer, then $q$ equals $\qquad$.
$$
a_{n}=n d, b_{n}=d^{2} q^{n-1} \Rightarrow \frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{b_{1}+b_{2}+b_{3}}=\frac{14}{1+q+q^{2}}=k \in \mathbf{N}^{*} \text{. }
$$

Let $q=\frac{s}{t}$, where $(s, t)=1$ and $t>s$. Then $\frac{s^{2}+s t+t^{2}}{t^{2}}=\frac{14}{k}$
$\Rightarrow\left(s^{2}+s t+t^{2}\right) k=14 t^{2} \Rightarrow t^{2} \mid k$. Let $k=m t^{2}$, then $s^{2}+s t+t^{2}=\frac{14}{m} \in \mathbf{N}^{*}$
$$
\Rightarrow s^{2}+s t+t^{2}=7 \Rightarrow s=1, t=2 \Rightarrow q=\frac{1}{2} .
$$