7. Let the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, where $a_{1}=p, b_{1}=q$. It is known that $a_{n}=p a_{n-1}, b_{n}=q a_{n-1}+r b_{n-1}(p, q, r$ are constants, and $q>0, p>r>0, n \geqslant 2)$, then the general term formula of the sequence $\left\{b_{n}\right\}$ is $b_{n}=$ $\qquad$
7. $\frac{q\left(p^{n}-r^{n}\right)}{p-r}$.

According to the problem, we have $a_{n}=p^{n}$.
$$
\begin{aligned}
b_{n} & =q a_{n-1}+r b_{n-1}=q p^{n-1}+r\left(q a_{n-2}+r b_{n-2}\right) \\
& =q p^{n-1}+r q p^{n-2}+r^{2} b_{n-2} \\
& =q\left(p^{n-1}+r p^{n-2}\right)+r^{2}\left(q a_{n-3}+r b_{n-3}\right) \\
& =q\left(p^{n-1}+r p^{n-2}+r^{2} p^{n-3}\right)+r^{3} b_{n-3} \\
& \cdots \cdots \\
& =q\left(p^{n-1}+r p^{n-2}+r^{2} p^{n-3}+\cdots+r^{n-2} p\right)+r^{n-1} b_{1} \\
& =q\left(p^{n-1}+r p^{n-2}+\cdots+r^{n-1}\right)=\frac{q\left(p^{n}-r^{n}\right)}{p-r} .
\end{aligned}
$$