Finding the number of iterations to a recurrence I have an algorithm where the number of items in my set decrease by $\sigma/(1+\sigma)$ on each iteration until all items are exhausted.

$$
\begin{align*}
S_0 &amp;= S \\
S_{k+1} &amp;= S_k - S_k \frac{\sigma}{1+\sigma}
\end{align*}
$$

Here $\sigma$ is a small value.

How can I find number of iterations? I know it is a geometric series but can't seem to simplify for number of iterations.
Presumably the correct form of the recursive step is
$$ S_{k+1} \leq S_k - S_k \frac{\sigma}{1+\sigma} = \frac{S_k}{1+\sigma}. $$
Using induction, we can show that
$$ S_t \leq \frac{S}{(1+\sigma)^t}. $$
So the algorithm will end by the time $T$ satisfying
$$ \frac{S}{(1+\sigma)^T} &lt; 1.$$
I'll let you solve this yourself.