Figurative Numbers

1. Write down the ﬁrst 10 square numbers. Why are they called square numbers?

2. If you build triangles out of dots and count the dots you get the triangular numbers. Here are the ﬁrst two triangular numbers represented geometrically (the ﬁrst is admittedly not much of a triangle):

Find the ﬁrst 10 triangular numbers.

3. Find a formula for the n-th triangular number.

4. What happens if you add two consecutive triangular numbers? Why does this happen? Can you prove your result geometrically? Can you prove your result algebraically?

5. Let T (n) represent the n-th triagular number. Write your previous results using this T (n) notation. (How do you write your general results for what you have discovered?)

6. If you add up triagular numbers you get the tetrahedral numbers. For example, the ﬁrst several tetrahedral numbers are:

1, 3, 6,...

Find the ﬁrst several tetrahedral numbers. Can you ﬁnd a formula for the n-tetrahedral number?

1 Figurative Numbers Part 2

1. Write down the ﬁrst 10 pentagonal numbers. To do this, draw pentagons with dots and increase the side length each time. Here are the ﬁrst few:

2. Write down the ﬁrst several hexagonal numbers.

3. Make a table for the first ten figurative numbers for the figures: Numbers 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th Triangular Square Pentagonal Hexagonal Heptagonal Octogonal Nonagonal Decagonal What patterns do you see?

2 Figurative Numbers Part 3

1. Can you ﬁnd a formula for P (n) (the n-th pentagonal number)?

2. Can you ﬁnd a formula for H(n) (hexagonal numbers)?

3. Explore centered ﬁgurative numbers. Here is a picture of what this means:

4. Given a triangular number, ﬁnd a formula to ﬁnd which triangular number it is. (This could be called the triangular root of a number, like a square root.)

5. (Fermat Polygonal Number Theorem)

(a) For each number 1 through 50, write it as a sum of triangular numbers. Try to use as few numbers as possible. (b) Try to write numbers are sums of square numbers. How many squares do you need? (c) Sums of pentagonal numbers? How many pentagonal numbers do you need?