Linear patterns
Identify and continue linear patterns from tables, find rules, and write equations to describe patterns and find specific terms.
Worked examples
Continuing a linear pattern
Straightforward
Problem
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 4 | 7 | 10 | 13 | ? |
Find the value at position 5.
1
Find the common difference.
, , . The value increases by 3 each time.
2
Add the common difference to the last known value.
Position 5:
Answer
The value at position 5 is .
Finding and using a pattern rule
Moderate
Problem
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 6 | 10 | 14 | 18 |
Find the rule and use it to calculate the value at position 12.
1
Find the common difference to determine in the rule .
, . The common difference is 4, so .
2
Substitute a known row to find .
When , :
3
Write the rule.
4
Substitute .
Answer
The rule is . The value at position 12 is .
Writing an equation and finding a term
Challenging
Problem
Fence posts are placed in a row. There are 3 posts in section 1, 5 posts in section 2, 7 posts in section 3, and the pattern continues.
Write an equation for the number of posts in section , and find how many posts are in section 30.
Write an equation for the number of posts in section , and find how many posts are in section 30.
1
List the values and find the common difference.
Section 1: 3, Section 2: 5, Section 3: 7. Common difference , so .
2
Use , to find .
3
Write the equation.
4
Verify with a known value.
: ✓
5
Substitute .
Answer
The equation is . Section 30 has posts.
Practise
Q1·Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 3 | 6 | 9 | 12 | ? |
What is the value at position 5?
Explanation
The value increases by 3 each time. After 12, the next value is .
Q2·Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 5 | 8 | 11 | 14 | ? |
What is the value at position 5?
Explanation
The value increases by 3 each time. After 14, the next value is .
Q3·Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 10 | 8 | 6 | 4 | ? |
What is the value at position 5?
Explanation
The value decreases by 2 each time. After 4, the next value is .
Q4·Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| Value | 4 | 7 | 10 | 13 | ? |
What is the value at position 6?
Explanation
The value increases by 3 each time. Position 5 gives , and position 6 gives .
Q5·Moderate
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 5 | 9 | 13 | 17 |
The rule is: Value for some constant . What is ?
Explanation
When , Value . So , giving . The rule is .
Q6·Moderate
A linear pattern follows the rule: Value .
What is the value when ?
What is the value when ?
Explanation
Value .
Q7·Moderate
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 7 | 12 | 17 | 22 |
Using the rule for this pattern, what is the value when ?
Explanation
The common difference is 5, so the rule has the form . When , Value : , so . The rule is . When : .
Q8·Moderate
A linear pattern follows the rule: Value .
At which position does the value equal 35?
At which position does the value equal 35?
Explanation
.
Q9·Challenging
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 8 | 13 | 18 | 23 |
Write the rule in the form and use it to find the value at position 20.
Explanation
The common difference is 5, so . When , : . The rule is . At : .
Q10·Challenging
A rule for a linear pattern is .
What is the first position at which the value exceeds 50?
What is the first position at which the value exceeds 50?
Explanation
. The first whole-number position is . Check: . ✓
Q11·Challenging
A linear pattern is described by the table below.
| Position () | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Value | 9 | 15 | 21 | 27 |
Write the rule in the form and find the value at position 15.
Explanation
From to , increases by 6 over 2 steps, so the common difference per unit is 3: . Using , : . The rule is . At : .
Q12·Challenging
Tiles are arranged in rows. Row 1 has 5 tiles, row 2 has 8 tiles, row 3 has 11 tiles, and the pattern continues.
How many tiles are in row 25?
How many tiles are in row 25?
Explanation
The common difference is 3, so . When , : . The rule is . At : .
Open Math
Linear patterns
Algebra · MA4-LIN-C-01
Name:
Date:
Q1Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 3 | 6 | 9 | 12 | ? |
What is the value at position 5?
Q2Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 5 | 8 | 11 | 14 | ? |
What is the value at position 5?
Q3Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 10 | 8 | 6 | 4 | ? |
What is the value at position 5?
Q4Straightforward
A pattern produces these values:
| Position () | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| Value | 4 | 7 | 10 | 13 | ? |
What is the value at position 6?
Q5Moderate
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 5 | 9 | 13 | 17 |
The rule is: Value for some constant . What is ?
Q6Moderate
A linear pattern follows the rule: Value .
What is the value when ?
What is the value when ?
Q7Moderate
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 7 | 12 | 17 | 22 |
Using the rule for this pattern, what is the value when ?
Q8Moderate
A linear pattern follows the rule: Value .
At which position does the value equal 35?
At which position does the value equal 35?
Q9Challenging
A linear pattern has the table below.
| Position () | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Value | 8 | 13 | 18 | 23 |
Write the rule in the form and use it to find the value at position 20.
Q10Challenging
A rule for a linear pattern is .
What is the first position at which the value exceeds 50?
What is the first position at which the value exceeds 50?
Q11Challenging
A linear pattern is described by the table below.
| Position () | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Value | 9 | 15 | 21 | 27 |
Write the rule in the form and find the value at position 15.
Q12Challenging
Tiles are arranged in rows. Row 1 has 5 tiles, row 2 has 8 tiles, row 3 has 11 tiles, and the pattern continues.
How many tiles are in row 25?
How many tiles are in row 25?
Worked solutions and answers at openmath.au/year-7/linear-relationships/linear-patterns