7. Given that $p$ and $q$ are coprime positive integers, and $n$ is a non-negative integer. How many different integers can be expressed in the form $i p + j q$, where $i$ and $j$ are non-negative integers, and $i + j \leqslant n$.
（Li Weiguo, Contributed)
7. If we denote $A(p, q, n)=|\dot{p}+j q| i, j \geqslant 0, i+j \leqslant n$, then the number of elements in $A(p, q, n)$ is
$$
|A(p, q, n)|=\left\{\begin{array}{ll}
\frac{(n+1)(n+2)}{2}, & n < p; \\
\frac{p(2 n-p+3)}{2}, & n \geqslant p.
\end{array}\right.
$$

According to the definition, we have
$$
\begin{array}{l}
A(p, q, n) \backslash A(p, q, n-1) \\
=\{i p+(n-i) q \mid i=1,2, \cdots, n\}.
\end{array}
$$

Notice that $i p+(n-i) q=p(i+q)+q(n-p-i)$, and
$(i+q)+(n-p-i)=n-p+q \leqslant n-1$, then
$$
i p+(n-i) q \in A(p, q, n-1) \Leftrightarrow n-p-i \geqslant 0.
$$

Therefore, $A(p, q, n) \backslash A(p, q, n-1)$
$$
=\left\{\begin{array}{l}
|\dot{p}+(n-i) q| i=n-p+1, n-p+2, \cdots, n \mid, n \geqslant p; \\
|\dot{p}+(n-i) q| i=0,1, \cdots, n \mid, n < p.
\end{array}\right.
$$

Let $a_{n}=|A(p, q, n)|$, then
$$
a_{n}-a_{n-1}=\left\{\begin{array}{cc}
p, & n \geqslant p; \\
n+1, & n < p.
\end{array}\right.
$$

Noting that $a_{0}=1$, for $n < p$, we have
$$
\begin{array}{l}
a_{n}=a_{0}+\left(a_{1}-a_{0}\right)+\cdots+\left(a_{n}-a_{n-1}\right) \\
=1+2+\cdots+(n+1)=\frac{(n+1)(n+2)}{2};
\end{array}
$$

Therefore, for $n \geqslant p$, we have
$$
\begin{array}{l}
a_{n}=a_{p-1}+\left(a_{p}-a_{p-1}\right)+\cdots+\left(a_{n}-a_{n-1}\right) \\
=a_{p-1}+(n-p+1) p=\frac{p(2 n-p+3)}{2}.
\end{array}
$$