TITLE: What is the nth term of the series? $0, 0 , 1 , 3 , 6 , 10 ....$
QUESTION [7 upvotes]: I am trying to find the relation between the number of nodes and the number of connections possible. So if there are $0$ nodes, that means $0$ connections possible, $1$ node still means $0$ connections possible, $2$ nodes $1$ connection possible, $3$ nodes $3$ connection and so on. How can I find the relation between $n$ nodes and the number of possible connections ?

REPLY [8 votes]: Let $a_n$ be the number of connections for $n$ nodes.
For $n > 0$, we have $a_n = a_{n-1} + n-1$ because among the $a_n$ connections, there are $n-1$ of them that connect to the $n^{th}$ node while $a_{n-1}$ of them didn't.
Rewrite above expression as $a_k - a_{k-1} = k-1$.
Start summing both sides from $k = 1$ to $n$, we get
$$a_n - a_0 = \sum_{k=1}^n (k-1) = \sum_{k=1}^n \frac{k(k-1)-(k-1)(k-2)}{2} = \frac{n(n-1)}{2}$$
Together with $a_0 = 0$, we find $a_n = \frac{n(n-1)}{2}$ for $n > 0$.
Since this is trivially true at $n = 0$, we have $a_n = \frac{n(n-1)}{2}$ in general.