Pascal’s Triangle Sequences part 1

Pascal's Triangle

Here we are again for Pascal’s Triangle Friday! I am pretty excited about this one today, because of how many things we are going to find when we look for sequences in Pascal’s triangle.

However, I’m going to start with telling you what I remember from being taught Pascal’s triangle at school: you add up the two numbers above to find the number below.

That’s it.

Dull, huh? That’s why I’m so keen that you find out how great Pascal’s triangle is!

How did I get from dull to delighted? Well, I had to teach a lesson on Pascal’s triangle for an interview. When I started to research it I was amazed! I had so much more than one lesson’s worth of material that I struggled to cut it down! You will see why as we look at some of the sequences today.

At the end of this post I have provided blank triangles which you are welcome to print out and use with your students (or for yourself) and you can fill it in using the method I mentioned above, which looks like this:

How to calculate Pascal's Triangle

That is the quick way to work out what the numbers in each section will be (also, notice that Pascal’s triangle is symmetrical, so you only need to work out one half).

So here is my filled-in triangle, and notice I have highlighted three diagonals.

Sequences in diagonals

Orange diagonal: all 1s. Which is a sequence 🙂
Yellow diagonal: the Natural Numbers. Or positive integers. Or counting numbers. Whatever you prefer to call them!
Pink diagonal: where it gets a little bit more interesting! This is the sequence of triangular numbers. “Why do you call them triangular numbers?”, I (imagine I) hear you ask. It isn’t because they’re in Pascal’s triangle, but because we can draw them as triangles:

Triangular Numbers
We aren’t going to stop there though! You know that if you put triangles together you can make squares? So here we go:

Triangular and Square Numbers
Square Numbers from Triangular NumbersSquare NumbersAdding two consecutive triangular numbers together gives you the square numbers!

So what did we find in Pascal’s triangle this week? The natural numbers; triangular numbers; and the square numbers. Come back next Friday for some more sequences from Pascal’s triangle!

Pascal’s Triangle series:

Pascal’s Triangle with Routes

Pascal’s Triangle with Probability

Pascal’s Triangle Sequences Part 1

Pascal’s Triangle Sequences Part 2

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