Triangular Day: More Figurative Families

Two weeks ago, we talked about the relationships between figurative families. We looked at these in a more basic light, by just analyzing the quick and easy patterns that are noticed. This week, I would like to go a little deeper.

Let's find the explicit formula for the different sequences. We'll start with triangular numbers.

T_n = n(n + 1)/2

We have used this formula in many of the previous posts about triangular numbers, so that one didn't take as much work. What about the square numbers?

S_n = n²

This formula is pretty obvious, considering that the definition of square numbers is a natural number squared.

What about pentagonal numbers? This might take some more work, but it can be found by solving a system of equations. At the end, you would get:

P_n = n(3n - 1)/2

What about hexagonal numbers? Again, this one would take some work. Let's see what it ends up with:

H_n = 2n² + n

There isn't an obvious pattern right now, but let's rewrite each thing in the terms of n(an - b)/2 with a and b being coefficients and constants in the formula.

T_n = n(1n - (-1))/2
S_n = n(2n - 0)/2
P_n = n(3n - 1)/2
H_n = n(4n - 2)/2

Now the pattern is pretty clear. Each formula is just adding one to a and b. Algebraically, a will always be equal to the number of sides on the figure minus 2, and b will be equal to the number of sides minus 4. I found this pattern to be pretty cool, considering how different the different shapes are.

Bonus:
I just found out that my TEDx talk in India was posted on YouTube. Here is the video of it.

I will also be speaking at TEDxBushnellPark in a week, and I will make sure to post that video when it is available as well.