Consider the following:
$$\sum_{k=0}^{n-1} v^k \mathbb{1}_{ \{K \geq k\}} \overset{!}{=} \frac{1-v^n}{1-v} $$
$K=0,1,2,3,.......$ is a random variable. I do not think that I can apply the geometric sum formula here. I do not know how many summands will be on the left side?
What do you think?