MathNet 2004 — Part 01 Question 191

Let $a_0, a_1, a_2, \dots, a_n, \dots$ be a sequence of numbers satisfying $(3 - a_{n+1}) \cdot (6 + a_n) = 18$, and $a_0 = 3$. Then $\sum_{i=0}^{n} \frac{1}{a_i}$ equals ________.