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
Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

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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Additional Activities

Real World Application of Triangular Numbers

Reminders

  • Triangular numbers are numbers that make up the sequence 1, 3, 6, 10, . . .. The nth triangular number in the sequence is the number of dots it would take to make an equilateral triangle with n dots on each side.
  • The formula for the nth triangular number is (n)(n + 1) / 2.

Problem:

A computer programmer is setting up a LAN network, such that each computer in the network has 1 wire that loops around to itself and 1 wire between each computer in the network. In other words, we have the following:

  • If there is 1 computer in the network, then 1 wire is needed (the wire that loops around from that computer to itself).
  • If there are 2 computers in the network, then 3 wires are needed (1 wire looped from each computer to itself, accounting for 2 wires, and 1 wire between the two computers in the network).
  • If there are 3 computers in the network, then 6 wires are needed (1 wire from each computer to itself, accounting for 3 wires, and 1 wire between each of the computers in the network, accounting for 3 wires).
  • If there are 4 computers in the network, then 10 wires are needed (1 wire from each computer to itself, accounting for 4 wires, and 1 wire between each of the computers in the network, accounting for 6 wires).

This pattern continues for each computer added to the network. Based on this network setup, answer the following questions.

Questions:

  1. Consider the sequence of numbers with the nth term being the number of wires needed for a network with n computers. What type of numbers make up the terms of this sequence?
  2. Based on the answer to the first question, what is the formula for the nth term in the sequence? In other words, how many wires would be needed for a LAN network described with n computers in it?
  3. How many wires would be needed if the network had 20 computers? How many wires would be needed if the network had 30 computers?
  4. A computer supply store sells wires, such that the cost is as follows:
  • 10 or less wires are $1 each.
  • 11-20 wires are $0.90 each.
  • 21-30 wires are $0.60 each.
  • 31-40 wires are $0.58 each.
  • 41-50 wires are $0.53 each.
  • 51 or more wires are $0.50 each.

Putting the cost of computers aside, would the cost of the wires required for a network be cheapest for 5 computers in the network, 6 computers in the network, or 7 computers in the network?

Solutions:

  1. Triangular numbers
  2. (n)(n + 1) / 2
  3. 20 computers would require 210 wires, and 30 computers would require 465 wires.
  4. 5 computers would require 15 wires, so they would cost $0.90 each. Thus, if a network had 5 computers, then the total cost for the wires would be 15 × $0.90 = $13.50. 6 computers would require 21 wires, so they would cost $0.60 each. Thus, if a network had 6 computers, then the total cost for the wires would be 21 × $0.60 = $12.60. Lastly, 7 computers would require 28 wires, so they would cost $0.60 each. Thus, if a network had 7 computers, then the total cost for wires would be 28 × $0.60 = $16.80. Of these costs, $12.60 is the least, and that happens when there are 6 computers in the network.

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