GCSE Maths Number

Types of Numbers

Triangular Numbers

# Triangular Numbers

Here we will learn about triangular numbers, including how to find the next triangular number in a sequence (including picture sequences). We will also learn how to find triangular numbers and determine whether a number is a triangular number using the nth term.

There are also triangular numbers worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are triangular numbers?

Triangular numbers are numbers that can be represented as a triangle. The numbers form a sequence known as the triangular numbers.

The first triangular number T_{1}=1 .

The second triangular number is found by adding 2 to the previous one.

So, T_{2}=1+2=3.

The third triangular number is found by adding 3 to the previous one.

So, T_{3}=3+3=6.

Continuing this pattern, you 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.

Visual representation

Triangular numbers can be represented using equilateral triangles. The number of dots within each triangle determines the value of the term. Each new row of dots in the triangle contains one more dot than the row above, creating a triangular pattern.

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.

To determine the next triangular number in a pictorial sequence, we add another row to the triangle that contains one more element than the previous row.

### What are triangular numbers? ### Nth term of the sequence of triangular numbers

The triangular numbers form a type of quadratic sequence. The nth term of the sequence is,

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

To calculate a term within the sequence of triangular numbers, we substitute the position of the term into the nth term (this would be the value of n ) and calculate the resulting value.

For example,

To find the 5th triangular number, n=5, so

T_{5}=\frac{1}{2} \times 5 \times(5+1)=15.

### Pascal’s triangle

Pascal’s triangle shows 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.

The triangular numbers 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.

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

## 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.

### Explain how to use triangular numbers ## 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.

6-3=3.

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.

As we want to calculate the next term, we can now draw the picture.

### Example 2: triangular number sequence

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

As we are given the first four triangular numbers, we can calculate the difference between the last two terms, and add one more than this value to get the next number in the sequence.

10-6=4

Adding one to the previous difference means that the next term will be 5 more than 10.

10+5=15

The next term will be 6 more than 15.

15+6=21

So, 21 is 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?

As we are given the first four triangular numbers, we can calculate the difference between the last two terms and add one more than this value to get the next term.

10-6=4

Adding one to the previous difference means that the next term will be 5 more than 10.

10+5=15

The next term will be 6 more than 15.

15+6=21

The next term will be 7 more than 21.

21+7=28

The next term will be 8 more than 28.

28+8=36

The next term will be 9 more than 36.

36+9=45

So, 45 is the 9th triangular number.

## How to calculate triangular numbers, using the nth term

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

1. Substitute the position of the term ( \bf{n} ) into the \bf{nth} term.
2. Complete the calculation.

### Explain how to calculate triangular numbers, using the nth term ### Example 4: finding a small value using the nth term

Calculate the 10th triangular number.

The nth term for the sequence of triangular numbers is

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

Here, we want to calculate the 10th term and so n=10.

Substituting n=10 into the nth term, you have

T_{10}=\frac{1}{2}\times{10}\times(10+1) .

\begin{aligned} T_{10}&=\frac{1}{2}\times{10}\times(10+1)\\\\ &=5\times{11}\\\\ &=55 \end{aligned}

The 10th term is 55.

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

Calculate the 50th triangular number.

The nth term for the sequence of triangular numbers is

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

Here, we want to calculate the 50th term and so n=50.

Substituting n=50 into the nth term, you have

T_{50}=\frac{1}{2}\times{50}\times(50+1) .

\begin{aligned} T_{50}&=\frac{1}{2}\times{50}\times(50+1)\\\\ &=25\times{51}\\\\ &=1275 \end{aligned}

So, the 50th term is 1275.

## How to work out if a number is a triangular number

In order to determine whether a number is a triangular number, using the nth term:

1. Form an equation where the nth term is equal to the value given.
2. Multiply both sides by \bf{2} .
3. Expand the brackets.
4. Rearrange the equation to get a quadratic in the form ax^{2}+bx+c=0 .
6. Write the solution.

### Explain how to work out if a number is a triangular number ### Example 6: triangular numbers using the nth term

Is 120 a triangular number?

The nth term of the triangular number sequence is

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

As we want to find if 120 is a triangular number, we can form the equation

\frac{1}{2}n(n+1)=120 .

n(n+1)=240

n^{2}+n=240

We need to subtract 240 from both sides of the equation so that the quadratic is equal to 0.

n^{2}+n-240=0

By solving the quadratic, we determine the position of the term within the sequence. If a solution is a positive integer, then 120 is a triangular number. If both solutions are not a positive integer, then 120 is not a triangular number.

n=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} .

For the quadratic n^{2}+n-240=0, \ a=1, \ b=1 and c=-240.

Substituting these values into the quadratic formula, we have

n=\frac{-1\pm\sqrt{1^{2}-4\times{1}\times{-240}}}{2\times{1}} .

Simplifying this, we get

n=\frac{-1\pm\sqrt{1--960}}{2}=\frac{-1\pm\sqrt{1+960}}{2}=\frac{-1\pm\sqrt{961}}{2} .

The square root of 961 is 31 and so we now have

n=\frac{-1\pm{31}}{2} .

So our solutions are n=\frac{-1+31}{2}=15 \ or \ n=\frac{-1-31}{2}=-16.

As one of the solutions is a positive integer (n=15) the number 120 is the 15th term in the sequence of triangular numbers, and so 120 is a triangular number.

### Common misconceptions

• The sequence is a linear sequence

The difference between two terms is considered to be common and so the next term is found by adding the difference from the previous two terms, not one more than the difference of the previous two terms.

• 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. (See examples 5 and 6).

### Practice triangular number questions

1. Below is a pattern sequence. Which picture shows the correct pattern for the term T_{5}?        The number of ovals increases by 1 for every new row above the triangle. The new row will contain 5 ovals, above the image for T_{4}. 2. What type of number is 28?

Square Number Decimal Number Cube Number Triangular Number The sequence of triangular numbers is

1, 3, 6, 10, 15, 21, 28, …

So, 28 is a triangular number.

3. What is the position of the term 36 in the sequence of triangular numbers?

9th term 8th term 36th term 6th term The sequence of triangular numbers is

1, 3, 6, 10, 15, 21, 28, 36, …

The position of the value 36 is the 8th term.

4. What is the value of the 12th triangular number?

78 72 144 66 Substituting n=12 into the nth term for the triangular numbers sequence, we have

T_{12}=\frac{1}{2}\times{12}\times{13}=78 .

5. What is the value of the 28th triangular number?

392 784 378 406 Substituting n=28 into the nth term for the triangular numbers sequence, we have

T_{28}=\frac{1}{2}\times{28}\times{29}=406 .

6. Which method correctly shows that 55 is a triangular number.        Forming an equation using the nth term of the sequence of triangular numbers, we have

\frac{1}{2}n(n+1)=55 .

Multiplying both sides of the equation by 2, we have

n(n+1)=110.

Rearranging the equation to form a quadratic of the form

ax^{2}+bx+c=0 we have

n^{2}+n-110=0 .

As we can factorise 110, we do not need to use the quadratic formula. We just need to be able to solve the quadratic equation we have formed.
(n-10)(n+11)=0

The two possible solutions are

n=10\text{ or }n=-11 .

The position in a sequence must be a positive integer, so n=10.

Therefore, 55 is the 10th triangular number.

### Triangular numbers GCSE questions

1. (a) The diagram below shows the number of handshakes made between an increasing number of guests at a party. Everyone shakes hands with all of the other guests.

Draw the next pattern in the sequence.

(b) How many handshakes are given between 6 people?

(4 marks)

(a) (1) (1)

(b)

0, 1, 3, 6, 10, 15, 21, …

(1)

15 handshakes

(1)

2. (a) Which of the following is the nth term of the sequence of triangular numbers?

(b) Show that the nth term of the sequence 1, 3, 6, 10, 15, … is \ \frac{1}{2}n(n+1).

(5 marks)

(a)

\frac{n}{2}(n+1)

(1)

(b)

Second difference =1 and so a=\frac{1}{2}.

(1)

\frac{1}{2}n^{2}=0.5, \ 2, \ 4.5, \ 8, \ 12.5, …

(1)

\frac{1}{2}n=0.5, \ 1, \ 1.5, \ 2, \ 2.5, …

(1)

\frac{1}{2}(n^{2}+n)=\frac{1}{2}n(n+1)

(1)

3. Show that the sum of any two triangular numbers is a square number.

(5 marks)

\frac{n}{2}(n+1) \ or \ \frac{n+1}{2}(n+2)

(1)

\frac{n}{2}(n+1) \ and \ \frac{n+1}{2}(n+2)

(1)

\frac{n}{2}(n+1)+\frac{n+1}{2}(n+2)=\frac{n^{2}+n}{2}+\frac{n^{2}+3n+2}{2}

(1)

=\frac{2n^{2}+4n+2}{2}=n^{2}+2n+1

(1)

=(n+1)^2

(1)

## Learning checklist

You have now learned how to:

• Recognise and use sequences of triangular numbers
• Use the nth term of a triangular numbers sequence to solve problems

## Still stuck?

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#### FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.