Triangular numbers

Here you will learn about triangular numbers, including how to identify them and work with them in numerical and pictorial sequences. You will also learn how to find triangular numbers and determine whether a number is a triangular number using the nth term.

Students first learn about triangular numbers when they learn about sequences and series in high school math classes such as Algebra I and Algebra II.

What are triangular numbers?

Triangular numbers are numbers that can be represented by a pattern of dots arranged in triangle (equilateral triangle) with the same number of dots on each side of the triangle. The number of dots form a sequence known as triangular numbers.

The total number of dots in each triangular pattern is the triangular number. Therefore, 1, \, 3, \, 6, and 10 are the first four triangular numbers.

If this pattern were to continue, how many dots would the side of the next triangle have? What would be the total number of dots the triangle has?

The next triangle is below. There are 5 dots that make up each side of the triangle and there are a total of 15 dots that make up the triangle.

So, the next triangular number is 15. Notice each of the row of dots starting from the bottom and moving to the top (5 dots, 4 dots, 3 dots, 2 dots, 1 dot).

Do you notice a pattern from triangle to triangle or from total amount of dots to total amount of dots?

From number to number in this sequence, you can draw the triangle with dots or look for another pattern. In this case, the pattern is +2, \, +3, \, +4, \, +5, etc.

Meaning, 1+2=3, \, 3+3=6, \, 6+4=10, \, 10+5=15

If you did not draw the next triangle in the sequence and used the pattern identified above, what do you think the total number of dots would be?

“21” would be the next triangular number because adding 6 to 15 is 21. If the triangle were to be drawn each side of the triangle would have 6 dots.

Continuing the pattern, you would get,

This produces the sequence of triangular numbers,

1, \, 3, \, 6, \, 10, \, 15, \, 21, \, 28, \, 36, \, 45, \, 55, \, 66, \, 78, \, 91, \, 105, \, 120, \, …

To determine the next triangular number in a numerical sequence, when given the sequence, we need to find the difference between the previous two terms and add one more than this value.

What if you were asked to find the 25 th triangular number in the sequence? Would you want to sketch out 25 “dot” triangles? The answer is “no.”

Sketching out 25 triangles is not an efficient way to find out the 25 th triangular number. However, there is a formula you can use to calculate any triangular number.

This formula is also known as finding the “nth” term of a sequence (or nth triangular number) where T_{n} is the number in the sequence you are looking for and “n” is the term in the sequence.

So, in this case, finding the 25 th term, you would make “n” equal to 25 to calculate it.

T_{n}=\cfrac{1}{2} \, n(n+1).

\begin{gathered}T_n=\cfrac{1}{2} \, (25)(25+1) \\\\ T_n=325 \end{gathered}

The 25 th term in the triangle number sequence is 325.

Here is some history on triangular numbers:

Triangular numbers are natural numbers. Carl Gauss and Pierre de Fermat are known for their work with number theory.

Triangular numbers were originally explored by the Pythagoreans who developed many relationships between different geometric shapes and numbers including triangular numbers, square numbers, pentagonal numbers (numbers represented within a regular pentagon) and hexagonal numbers (numbers represented within a regular hexagon).

These are sometimes known as polygonal numbers or figurate numbers.

Let’s take a look at another example of where triangular numbers show up – Pascal’s triangle.

Pascal’s triangle shows a pattern of numbers that represent the binomial coefficients of a binomial expansion (x+y)^{n} where the value in the row below is the sum of the two values above it.

You can identify the triangular numbers in Pascal’s triangle. They appear in Pascal’s triangle next to the natural numbers. This means that if we add these two values together, we will obtain the next triangular number below it.

Fun fact: Did you know: given two consecutive triangular numbers, the sum of triangular numbers is a square number!

What are triangular numbers?

Common Core State Standards

How does this relate to high school math?

• High School Functions: Interpreting Functions (HSF-IF.A.3):
  Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0)=f(1)=1, \, f(n+1)=f(n)+f(n-1) for n \geq 1

• High School Functions: Linear, Quadratic, and Exponential Models: (HSF-LE.A.2)
  Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

How to use triangular numbers

In order to calculate a triangular number, given a sequence:

1. Calculate the difference between two consecutive terms.
2. Add one more than this value to determine the next term.
3. Continue until you reach the required term in the sequence.

Triangular numbers examples

Example 1: triangular numbers picture sequence

Draw the next picture in the sequence, T_{4}.

1. Calculate the difference between two consecutive terms.

Term 2 contains 3 squares.

Term 3 contains 6 squares.

2Add one more than this value to determine the next term.

Adding one to the previous difference means that the next term will have another row added below containing 3+1=4 squares.

3Continue until you reach the required term in the sequence.

Continue the pattern by drawing four squares in the last row.

Example 2: triangular number sequence

The first four triangular numbers are 1, \, 3, \, 6, and 10. Is 21 a triangular number?

Example 3: triangular numbers – find the position

The first four triangular numbers are 1, \, 3, \, 6, and 10. What position in the sequence is the triangular number 45?

How to calculate triangular numbers, using the nth term

In order to calculate a triangular number, using the nth term:

1. Identify the given information.
2. Substitute the values given into the equation.
   \hspace{4cm} T_n=\cfrac{1}{2} \, n(n+1)
3. Complete the calculation.

Example 4: finding a small value using the nth term

Calculate the 10 th triangular number.

Example 5: triangular numbers – large value using the nth term

Calculate the 50 th triangular number.

Example 6: triangular numbers using the nth term

What term or position is the triangular number, 105, in?

Teaching tips for triangular numbers

• Infuse learning tasks where students have opportunities to explore or investigate the triangular numbers so that they can make conjecture about the patterns.

• Have students explore other popular sequences such as the Fibonacci sequence and look to see where it might be represented in nature.

• Have struggling students use digital platforms to get online tutorials and extra practice.

• Instead of worksheets, use game-playing as a way to have students practice skills.

Easy mistakes to make

• Confusing the position and the value of the term
  The position of the term is the value of n. The value of the term is what the nth term is equal to, given the position n. This is the difference between substituting n into the nth term, or calculating the value of n using the nth term.

• Forgetting strategies for solving quadratic equations
  For example when solving for n or the position of a triangular number in the sequence requires solving a quadratic equation. Factoring or the quadratic formula are both good strategies to utilize when solving.

Related number patterns lessons

• Recursive formula
• Explicit formula
• Arithmetic sequence
• Geometric sequence formula
• Quadratic sequences
• Input/output tables

The next lessons are

• Functions in algebra
• Laws of exponents
• Scientific notation

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