4. Given an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference of $d \neq 0$, and $a_{1} 、 a_{3} 、 a_{9}$ form a geometric sequence, the value of $\frac{a_{1}+a_{3}+a_{9}}{a_{2}+a_{4}+a_{10}}$ is $\qquad$ .
4. $\frac{13}{16}$.

Let $a_{n}=a_{1}+(n-1) d$, so $a_{1}\left(a_{1}+8 d\right)=\left(a_{1}+2 d\right)^{2}$, which gives $a_{1}=d$, hence $a_{n}=n d$.
$$
a_{1}+a_{3}+a_{9}=13 d, a_{1}+a_{4}+a_{10}=16 d \text {, thus } \frac{a_{1}+a_{3}+a_{9}}{a_{2}+a_{4}+a_{10}}=\frac{13}{16} \text {. }
$$