I am looking at the following problem: suppose there are 20 people in line waiting to be served a food item. The server seeks to equally divide the amount of food available among the number of people in line he observes at the time of serving. Suppose there is only one tray of food to serve. Then he will serve the first person $1/20$ of the tray. However, as soon as the first person leaves the line, 20 new people join the line, so the server seeks to equally distribute the remaining $19/20$ of the tray among the now 39 people. This process continues so that every time someone leaves the line, 20 new people join the line and the server will serve the next person the remainder of the tray equally divided among everyone in line.
I am trying to find a formula (in terms of $n$) for the portion of the tray of food that the $n^{th}$ person in line will receive. I was able to write the following recursive formula for the amount, $a_n$, the $n$th person would receive:
$$a_n=\frac{1-\sum_{i=1}^{n-1}a_i}{19n+1}, \; a_1=1/20$$
However, I am trying to find a non-recursive expression that depends only on $n$.
I solved the problem for the easier case when there are initially 2 people (as opposed to 20) and 2 new people in the line each time someone is served: $a_n=\frac{1}{(n+1)n}=\frac{1}{2\sum_{i=1}^{n}i}$.
This second expression is revealing and I think there is an elegant direct interpretation of it; in effect, the $n^{th}$ person is getting half of the tray divided among 1 more than the sum of the line numbers before her turn ($2+3...+n$ i.e. 2 in line when the first served, 3 in line when second served, etc). I think the ``half" part relates to the fact that the first person only gets half the tray since there is another one in line at the time.
Is there such a direct interpretation that can produce the formula when there are 20 people?