Customers arrive at a restaurant according to a Poisson process with arrival rate $\lambda > 0$. As the head of the advertising agency for this restaurant, you are paid a comission of $i$ dollars for the $i$th customer. You are paid at time $T > 0$. What is the expected number of dollars in your paycheck?
My work, although I have low confidence it is correct:
Let $\{N(t)\}_{t=0}^{\infty} \sim PP(\lambda)$ be the number of arrivals at the restaurant. Let $S(t)$ be the amount of dollars in the paycheck at time $t$, then $S(t) = \sum_{i=1}^{N(t)}i = N(t)\cdot (N(t) + 1)/2$ by summation properties.
Condition on the number of arrivals by time $T$. Then, $E[S] = E[E[S|N(T) = k]]$. So,
$$ E[S] = \sum_{\text{all } k}E[S|N(T)=k]\frac{(\lambda T)^{k}e^{-\lambda T}}{k!}dk = \sum_{\text{all } k}\frac{k^{2} + k}{2}\frac{(\lambda T)^{k}e^{-\lambda T}}{k!} dk$$
There are problems here, I am sure, so I am looking for criticism and also a solution.