# Math Portfolio - Triangular Numbers

MOCK MATH PORTFOLIO

Triangular Numbers

1               3                          6                                  10

In a sequence of triangular numbers, the dots (as shown in the diagram above) represent a value of 1. In the sequence, the following triangle always has one more row of dots than the previous, and the first triangle has only one row of one dot. Each new last row in the next triangle always has one more dot than the last row in the previous triangle.

Since the first triangular number’s last row contains 1 dot, the next triangular number’s (the second triangular number’s) last row would contain 2 dots, and continue as such, and the number of any particular triangle in the sequence (the n value) would represent the number of dots in the last row of that triangle. Thus, in a continuation of the diagram, the fifth triangular number would be equivalent to the fourth triangle with an addition of a fifth row which contains five dots.

To complete the triangular number sequence with three more terms, the next three terms would be the 5th, 6th, and 7th triangular numbers in the sequence. Thus, they would be equivalent to the following:

5th number: (4th triangular number) + 5

= 10 + 5

= 15

6th number: (5th triangular number) + 6

= 15 + 6

= 21

7th number: (6th triangular number) + 7

= 21 + 7

= 28

Let the triangular number be.

This method of calculating the values of can be proven to be accurate using the method of differences. Here, the difference between consecutive values of   are calculated, and the data collected from that is manipulated in a manner to prove that the earlier used method of calculating subsequent values of  is reliable.

In this case, the second difference – the difference between any two consecutive values of the first difference (the difference between consecutive values of ) is calculated, and used to prove that the earlier used method of calculating subsequent values of  is reliable.

The sequence has a constant second difference of 1.

∴

∴

∴

When applied to the 5th, 6th, and 7th triangular numbers which hold n values of 5, 6, and 7 accordingly, the equation proves true:

To find, set  =

,    n = 4

, ,   →   ...