Eccoci.

Given is the arithmetic-geometric sequence $u(k) = (900-3k)r^{k - 1}$.
Let $s(n) = \sum_{k = 1}^n u(k)$.

Find the value of $r$ for which $s(5000) = -600\,000\,000\,000$.

Give your answer rounded to $12$ places behind the decimal point.