A ** polygonal
number** is a number that represents
the amount of dots that can be arranged evenly in the shape of
a regular polygon.

Pictures and formulas for triangular and square
numbers are given below. Also, pictures of pentagonal
and hexagonal numbers are given.
After you have viewed this page, go on to a discussion of general
**K-gonal** **numbers**.

Below, are the first five **triangular **numbers**.**
Notice that you begin with **1** and **add 2 **to
get to the second triangular number, **3.** Then you **add 3** to
get to the third triangular number, **6.** Then you **add 4** to get the fourth triangular, **10. **Then you **add
5** to get the fifth
triangular, **15.** Do you see the pattern?
It appears that you add the first **n** counting numbers (positive
integers) to get the **nth** triangular number. Can you predict
the 107th triangular number? Sure, **1 + 2 + 3 + ... + 107**.
Perhaps it would be nice to have an expression that produces any
triangular number.

1 3 6 10 15

Let **Tn **= the **nth** triangular
number. Then **Tn
= 1 + 2 + 3 + ... + n.** If you think
for a moment like Gauss (who, according to legend, found a quick
and clever way to add the first 100 counting numbers when his
teacher tried to keep him busy for a while), you should proceed
as follows:

Tn = 1 + 2 + 3 + ... + (n-2) + (n-1) + n+ Tn = n + (n-1) + (n-2) + ... + 3 + 2 + 1----------------------------------------------------2Tn = (n+1) + (n+1) + (n+1) + ... + (n+1) + (n+1) + (n+1) :nsets of(n+1)2Tn = n(n+1)

.

Below, are the first five **square **numbers**.**
the pattern is a lot easier to see. The **nth** square number
is **n*n** = **n-squared**.

If you must feel sophisticated, you can also
follow the same reasoning as above. Notice that you begin with
**1** and **add 3 **to get to the second triangular
number, **4.** Then you **add
5** to get to the third triangular number,
**9.** Then you **add
7** to get the fourth
triangular, **16. **Then you **add 9** to get the fifth triangular, **25.** Do you see the pattern? It appears that you add the
first **n** odd positive integers to get the **nth** square
number. The **nth** odd number is **(2n-1)**.

1 4 9 16 25

Let **Sn **= the **nth** square
number. Then **Sn
= 1 + 3 + 5 + ... + (2n-1).**

Sn = 1 + 3 + 5 + ... + (2n-5) + (2n-3) + (2n-1)+ Sn = (2n-1) + (2n-3) + (2n-5) + ... + 5 + 3 + 1-------------------------------------------------------------------2Sn = 2n + 2n + 2n + ... + 2n + 2n + 2n :nsets of2n2Sn = n(2n)

.

But what is the pattern and formula for the
rest of the polygonal numbers??? In other words, how can I determine
the value of the **nth** polygonal number where the polygon
formed has **K** sides?

Let **Kn = **the **nth k-gonal**
number. What is the formula for **Kn**, a general **K-gonal number**?

**Pentagonal** numbers:

1 5 12 22 35

**Hexagonal **numbers:

1 6 15 28 45

top of page