The expression I have is the following:
$S = \sum_{i=1}^T (1-q)^{i-1}i$
My expression after I play around with it is:
$S= \frac{1-(1-q)^TT}{q}+\frac{1-q}{q} \left(\frac{1-(1-q)^{T-1}}{q} \right) $
This looks a bit too complex to me so I think I might have made a mistake. Maybe some detail is warranted. $q \in [0,1]$ and $T \in N$