An Arithmetic Geometric Sequence

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

Write-up coming later

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