What Are Triangular Numbers? - Definition, Formula & Examples

What Are Triangular Numbers? - Definition, Formula & Examples
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  • 0:04 Definition of…
  • 0:35 Counting Sequence
  • 2:43 Triangular Number Formula
  • 4:33 The 19th Triangular Number
  • 5:17 Lesson Summary
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Lesson Transcript
Instructor: Kimberly Osborn
Triangular numbers are used to describe the pattern of dots that form larger and larger triangles. This lesson will explore the rule behind this pattern and how it can be applied to find any term in the sequence.

Definition of Triangular Numbers

Equilateral triangles made from dots

Triangular numbers, as shown in the image here, are a pattern of numbers that form equilateral triangles. Each subsequent number in the sequence adds a new row of dots to the triangle.

It is important to note that in this case, n equals the term in the sequence. So, n equals 1 is the first term, n equals 5 is the fifth term, n equals 256 is the 256th term. Believe it or not, we can actually use this n to figure out how many dots are in its corresponding triangle (i.e. its triangular number).

Counting Sequence

Before we piece together the formula used to find any term in our pattern, let's first see if we can find a counting pattern in our first four terms. To do this, let's look at the number of dots in each row here:

Chart for triangular numbers

If we continue with this pattern, what would our fifth term look like?

n = 5: 1 + 2 + 3 + 4 + 5

Now, you might have noticed a relationship between the term and the last number in our counting sequence. They are the same number! You might have also noticed that each addition counts backwards from there.

Let's use this logic then, to look at our fourth term:

n = 4: 1 + 2 + 3 + 4

We know that the last number in our counting sequence is four. Starting with our four and counting backwards, we can rewrite this sequence as:

n = 4: (4 - 3) + (4 - 2) + (4 - 1) + 4

Taking it one step further, we can apply this to any term in our sequence. This means that:

nth term = 1 + 2 + 3 + … + (n - 3) + (n - 2) + (n - 1) + n

If you are anything like me, you might prefer to write this sequence the other way around so that you can easily apply it to other triangular numbers.

nth term = n + (n - 1) + (n - 2) + (n - 3) + … + 3 + 2 + 1

Let's try applying this counting sequence to the eighth triangular number.

n = 8: 8 + (8 - 1) + (8 - 2) + (8 - 3) + (8 - 4) + (8 - 5) + (8 - 6) + (8 - 7) + (8 - 8)

n = 8: 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0

When we add these all together, our eighth triangular number is 36. This means that the eighth equilateral triangle in the sequence has 36 dots.

While this is a simple way to find a triangular number, it can become difficult when you are working with larger numbers because it increases the amount we must add together. In this case, we have to use a formula.

Triangular Number Formula

In order to find the triangular numbers formula, we must first double the number of dots in each equilateral triangle to create a rectangle. While the reasoning behind this might not be clear just yet, you will definitely understand our need for a rectangle a little later in this section.

Rectangles created from doubling the triangles.

In the image here, our original triangles are in red with its double in purple to create the rectangle. Thinking back to our counting sequence, we remember that the term, n, is the same as the longest row of dots. In the image here, we can see that one side of the rectangle is labeled as our n dots, while the other side is labeled as n + 1 dots.

Let's take the third term, for example; the third equilateral triangle has one side of three dots and the other of (3 + 1) or four dots.

For our formula, we are going to use the variable T to define our triangular number (or the number of dots in each equilateral triangle.) This is what the formula is helping us solve for.

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