Triangular Day... times nine plus one

Today is another triangular day! It is July 28th, and 28 is the seventh triangular number.

Before we start, let's remind ourselves of the explicit formula (you plug in n and receive the nth term in the sequence) for triangular numbers. Considering we will be proving patterns, it will definitely come in handy.

T(n) = n(n + 1)/2

We'll get back to that later. In the meanwhile, let's look at this pattern:

Add one to any triangular number multiplied by nine and you have a new triangular number.

This seems a little crazy; is this sequence really that cool? Let's check a few values.

1) 1 x 9 + 1 = 10
2) 3 x 9 + 1 = 28
3) 6 x 9 + 1 = 55
4) 10 x 9 + 1 = 91
5) 15 x 9 + 1 = 136

It is working. Those are all triangular numbers. Even that 136 is in fact sixteen times seventeen over two. But why?

First off, let's look for a pattern in the outcomes.

The first triangular number gives us the fourth
The second gives us the seventh
The third gives us the tenth
The fourth gives us the thirteenth
The fifth gives us the sixteenth

Basically, we are just adding three. To make it more mathematical (or complicated), the formula for the output of this pattern while putting in the nth triangular number is:

(3n + 1)(3n + 2)/2

That is a little confusing, but we aren't looking for the nth triangular number. We are looking for the 3n + 1st triangular number, the output of the pattern, which can be solved by this formula.

However, we are really doing 9 times T(n) plus one, which would look like this:

9n(n + 1)/2 + 1

We can simplify that to:

[9n(n + 1) + 2]/2
(9n^2 + 9n + 2)/2

If we also simplify the formula above, we get:

(3n + 1)(3n + 2)/2
(9n^2 + 6n + 3n + 2)/2
(9n^2 + 9n + 2)/2

They are both equal! And there is your proof.

I normally post the algebraic proof of things like this mainly because it is difficult to put shapes in a blog post. However, triangular numbers also have some very elegant geometric proofs that actually are like puzzles; fitting nine triangles together into another triangle with just one dot missing, or any other pattern you like.

Answer: On June 30th, I posted a problem that we learned at CTY. Let me write it down for you again.

By pure random guessing, what is the probability that you will get this answer correct.
  a. 50%
  b. 25%
  c. 0%
  d. 50%

This is a little confusing to follow, but let's think about it. With four choices, there is a 25% chance that you will get it correct. Therefore, it is obviously b: 25%.

However, what are the odds that you choose a or d? 50%, correct? So if one of those two were an answer, you would have a 50% chance of choosing it meaning those are correct as well.

Yet, this gives you a 75% chance of getting the answer correct. The probability that you will choose 75% is 0%, considering that it is not an option. This makes the correct answer 0%, since there is no way you would actually get the correct answer.

Round and round we go. There is a case for every single answer on the board, meaning all of the answers are correct. Personally, I think c has the best, most in depth case. However, which ever answer you thought when I gave the problem is correct. Good job!