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}= n

^{2}

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}= 2n

^{2}+ 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.

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