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.

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

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

### 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 ${\displaystyle n}$  nodes has ${\displaystyle T_{n-1}}$  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.[1] How many routes would be required to allow for a direct flight between any two of these airports? (Answer, ${\displaystyle T_{5079}}$  = 12,900,660 routes).

What network topologies do the airlines use to reduce the number of individual routes while allowing efficient travel between any two airports. (Answer, airlines often use a hub-and spoke network architecture.)

### Pascal’s Triangle

${\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\\\end{array}}}$
Pascal's triangle with rows 0 through 7.

In Pascal’s triangle, shown at the right, each number is the sum of the two numbers directly above it. Notice the diagonals containing the sequence 1, 3, 6, 10, 15, 21, …. These diagonals are formed from the triangular numbers.