# What is the sequence of the diagonals of a polygon?

Apr 20, 2017

$\left\{\begin{matrix}{a}_{1} = 0 \\ {a}_{2} = 0 \\ {a}_{n} = \frac{n \left(n - 3\right)}{2} \text{ for } n \ge 3\end{matrix}\right.$

#### Explanation:

I think you are wanting to know what the number of diagonals of a convex $n$ sided polygon for any $n$, with the resulting sequence.

A diagonal runs from one vertex to a distinct non-adjacent vertex, of which there are $n - 3$ possibilities.

Hence the formula for the number of diagonals of an $n$-sided convex polynomial is:

$\frac{n \left(n - 3\right)}{2}$

which is valid for any $n \ge 3$

So we can define a corresponding sequence by:

$\left\{\begin{matrix}{a}_{1} = 0 \\ {a}_{2} = 0 \\ {a}_{n} = \frac{n \left(n - 3\right)}{2} \text{ for } n \ge 3\end{matrix}\right.$

The first few terms are:

$0 , 0 , 0 , 2 , 5 , 9 , 14 , 20 , \ldots$