This activity introduces the triangular numbers and explores various applications.

In the image on the right notice the spheres are arranged to form an equilateral triangle.

Here is a tabulation of the number of objects in each row, and the total number of objects.

Triangular Numbers
Row Number Number of objects in this row Total Number of Objects
1 1 1
2 2 3
3 3 6
4 4 10

The last column corresponds to the triangular numbers.

Extend this sequence. (Hint, the number of objects in each row equals the row number.)

(Answer: The sequence of triangular numbers begins: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136…)

## Analysis

The nth triangular number is given by the following formulas.

$T_{n}=\sum _{k=1}^{n}k=1+2+3+\dotsb +n={\frac {n(n+1)}{2}}$

What is the 10th triangular number? (Answer 10(10+1)/2 = 55. Note the sequence shown above begins with $T_{0}$ )

### Fully Connected Networks

In a fully connected network every node has a direct link to very other node. The figure on the right illustrates a fully connected network with 6 nodes. Note that it includes 15 connections. In general, a fully connected network with $n$  nodes has $T_{n-1}$  connections. This is analogous to counting the number of handshakes if each person in a room with n people shakes hands once with each person.
There are approximately 5080 public airports in the United States. How many routes would be required to allow for a direct flight between any two of these airports? (Answer, $T_{5079}$  = 12,900,660 routes).
${\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\\\end{array}}$