Math Adventures/Triangular numbers



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.

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).

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
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.