Finding the General Formula for the kth Polygonal Number

If we represent numbers using “dots,” there are special numbers that can form “polygons.” Numbers that can form polygons are called polygonal numbers. For example, the square numbers 1, 4, 9, 16 and so on form a “square.”  As shown below, it is easy to see that the 10th square number is a square with 10 rows and 10 columns. This means that the 10th square number is equal to 100 (has 100 dots).

The Representation of Square Numbers

Like square numbers, triangular numbers form a shape. You guessed it right, the shape is a triangle. Finding the 15th triangular number seems hard, but we have already learned that triangular numbers are connected to the sum of the first m positive integers. The mth triangular number has

$1 + 2 + 3 + \cdots + m = \frac{1}{2}m(m+1)$ dots. » Read more

Triangular Numbers and the Sum of the First n Positive Integers

The numbers 1, 3, 6, 10, 15, … are called triangular numbers because they could be arranged in the form of triangles. Triangular numbers is one of the polygonal numbers — numbers that can represented by dots to form regular polygons.

Finding the nth triangular number is quite easy. All we have to do is form a rectangle using the “dot representation” of two triangular numbers. For example, we want to find the fourth triangular number, we create dots representing two triangular numbers, and then use them to form a rectangle. The area of the formed rectangle is $4(4+1)$. » Read more

The 12 Days of Christmaths and the Triangular Numbers

Many of us are familiar with the Twelve Days of Christmas — a song that enumerates a series of increasingly grand gifts given on each of the twelve days of Christmas.  For those who are not familiar with the song, here are a few stanzas (for full lyrics, click here).

On the first day of Christmas,
my true love sent to me
A partridge in a pear tree.

On the second day of Christmas,
my true love sent to me
Two turtle doves,
And a partridge in a pear tree.

On the third day of Christmas,
my true love sent to me
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

On the fourth day of Christmas,
my true love sent to me
Four calling birds,
Three French hens,
Two turtle doves,
And a partridge in a pear tree.

The math: How many gifts was  given on the twelfth day? » Read more