7. If the sequence of positive numbers $\left\{a_{n}\right\}$ is a geometric sequence, and $a_{2} \cdot a_{6}+2 a_{4} a_{5}+a_{1} a_{9}=25$, then the value of $a_{4}+a_{5}$ is $\qquad$ .
7. 5 .

Since $a_{2} \cdot a_{6}=a_{4}^{2}, a_{1} \cdot a_{9}=a_{5}^{2}$, we get $a_{4}^{2}+2 a_{4} a_{5}+a_{5}^{2}=25$, thus $\left(a_{4}+a_{5}\right)^{2}=25$, which means $a_{4}+a_{5}=5$.