Patterns

Co-ordinates

Graphs

Pattern 1

Pattern 2

Pattern 3

Pattern 4

Rule for a linear pattern

When describing a pattern it is often possible to describe it using words. The issue with this is there are many different ways for a pattern to be describe and sometimes they can be lengthy. However, mathematicians are lazy and want to simplify the amount that is written. Below is the rule that can be used to describe any linear pattern.

General rule for a linear pattern

T is the value for the nth term in the pattern

a is the change between each term

n is what number in the pattern it is

b is the 'zero term' or the adjustment factor.

Find the difference between the two terms (make sure that it is the same between each).
Go back one term from the first term of your pattern to the 'zero term'

This is your 'a' in the equation
This is your 'b' in the equation

Example #1

3, 5, 7, 9, 11, ...

The difference between each term is +2 therefore our 'a' is 2
Going back one term from the first term would be 1. Meaning our zero term or 'b' is 1.

Therefore the rule becomes:

To interpret this pattern in words it would be that for each term you increase one term ('n') you increase by 2, however it needs to be adjusted by adding one.

Example #2

10, 8, 6, 4, 2, ...

The difference between each term is -2 therefore our 'a' is -2
Going back one term from the first term would be 12. Meaning our zero term or 'b' is 12.

Therefore the rule becomes:

Using a linear rule

The benefit of having a rule for a pattern is that you are then able to find any number in that pattern.

For example if you were asked to find the 15th number a pattern the rule makes it easy to without having to list out every term out to the 15th term. The term is found by simply substituting into the equation.

Example

Q: Use the rule to calculate the 11th term in this pattern

Remember: In algebra if there is no operation sign between a number and a letter it means multiplied

Since we are looking for the 11th term we replace n (which represents the term number) with 11

Then we complete the calculation using BEDMAS to come to our answer of 47.

This means the value of the 11th term is 47

Finding a term number

Ocassionally you might receive a question where you are given a number in the pattern and you are required to find out where it is in the sequence.

To do this 'T' is placed for the term number, and solved using the term number to find 'n'

Patterns

Co-ordinates

Graphs

Pattern 1

Pattern 2

Pattern 3

Pattern 4

Rule for a linear pattern

General rule for a linear pattern

T is the value for the nth term in the pattern

a is the change between each term

n is what number in the pattern it is

b is the 'zero term' or the adjustment factor.

Example #1

3, 5, 7, 9, 11, ...

Example #2

10, 8, 6, 4, 2, ...

Using a linear rule

Example

Q: Use the rule to calculate the 11th term in this pattern

Finding a term number

Example

Q: Use the rule to what term number has the number 86.

Using the rule:

Substitute in 86

Subtract 1 From each side

Divide both sides by 5

The 17th term has the value 76.