Summation formula for basic math question I am reviewing some old retired case studies we have at work that we used to give to candidates interviewing. There is a short question on one part of the case that is pretty straight forward, but I am curious on an 'easy' way to solve an additional question that could have been asked. 
Question: You start with 1M customers. You grow by 1M customers each month (so month 1 is 1M customers). 40% of 1M customers each month will pay you 5 dollars (monthly subscription fee for a certain product). What do we make in the first month? So.. 1M * .40 * $5 -> $2M, 400k incremental customers each month.
What if the question asked how much do we make at the end of the year? The first month we have 400k customers that pay $5, but the second month we have 400k + new 400k that pay 5 dollars. All the way to month 12. You can of course do this manually but that would take a while, and it is easy to do in Excel.. but is there a summation formula that can be used here?
 A: It seems like $40\%$ of your customers in month $n$ pay you $\$5$ each. In month $n$, you have $n$ times as many customers as in month $1$, so receive $n$ times as much money as in month 1 ($n=1,2,\ldots,12$). So in month $n$, you are receiving $2000000n$ dollars. The total money received in dollars in the first $12$ months is thus
$$\sum\limits_{n=1}^{12}2000000n = 2000000\sum\limits_{n=1}^{12}n.$$
Then using the fact that $\sum\limits_{n=1}^Nn= N(N+1)/2$ for any positive integer $N$, we have $\sum\limits_{n=1}^{12}n = 12\times 13/2 = 78$. Hence you receive $$2000000\times 78 = 156 \text{ million dollars}.$$

In summary, yes, there is a summation formula that can be used. The key ingredient is the formula $$\sum\limits_{n=1}^Nn= N(N+1)/2$$ (i.e. $1+2+\cdots+N = N(N+1)/2$).
In this case, $N$ is quite small ($12$), so candidates who didn't know the formula could probably manually calculate $1+2+\cdots + 12$ still. (So if you wanted to reduce the chances of candidates manually calculating, you could change $N$ from $12$ to a bigger number, like $24$ or $36$.)
