Unified weak Goldbach conjecture for prime-started arithmetic progressions

Let $A(p,q)$ be the arithmetic progression whose first two terms are odd primes $p$ and $q$, and let $n$ be a positive integer. Unified weak Goldbach conjecture. For each positive integer $a>2$, there exist a positive integer $n$ and odd primes $p$ and $q$ such that

$$a=\frac{n}{2}((n-1)q-(n-3)p).$$

The source states that this formulation holds if and only if the even and odd weak Goldbach conjectures stated earlier both hold. It packages the even- and odd-length initial-sum representations into a single assertion, and no proof or resolution is supplied.