Taking $ n = 1, $ we get $ pq + r = 14 $. Also, from the formula $ a_n = 24 - 5a_{n - 1}, $ \[p \cdot q^n + r = 24 - 5(p \cdot q^{n - 1} + r) = 24 - 5p \cdot q^{n - 1} - 5r.\]We can write this as \[pq \cdot q^{n - 1} + r = 24 - 5p \cdot q^{n - 1} - 5r.\]Then we must have $ pq = -5p $ and $ r = 24 - 5r $. Hence, $ 6r = 24, $ so $ r = 4 $. From $ pq + 5p = 0, $ $ p(q + 5) = 0, $ so $ p = 0 $ or $ q = -5 $. If $ p = 0, $ then $ r = 14, $ contradiction, so $ q = -5 $. Then \[-5p + 4 = 14,\]whence $ p = -2 $. Therefore, $ p + q + r = (-2) + (-5) + 4 = \boxed{-3} $.