Spring 2001

Essay 2

Polygonal Numbers

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)   : n sets 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   : n sets of 2n
2Sn = 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?

OTHER ESSAYS

Pentagonal numbers:

`                            `
`        1           5              12                  22                       35       `

Hexagonal numbers:

`                            `
`        1           6              15                  28                      45`

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OTHER ESSAYS