Linear sequences

Gold-standard guide

20 mins

What you will learn

Find patterns, nth terms and missing values.

Use a clear step-by-step method for linear sequences.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

The common difference d is the step between consecutive terms

Step 1

Find the common difference

From term 3 to term 8 is 5 steps

Step 2

Use the nth term form: T(n) = dn + c = 4n + c

We know T(3) = 13, so 4(3) + c = 13

Step 3

Write the nth term

T(n) = 4n + 1

Watch out

Students confuse the common difference with the nth term itself

f

Linear nth term

$nth term = difference \times n + adjustment.$

f

Check

$Substitute n = 1 and n = 2 to check the first terms.$

Worked example

A linear sequence has the 3rd term 13 and the 8th term 33. Find the nth term formula.

1

Find the common difference: From term 3 to term 8 is 5 steps. The difference in values is 33 − 13 = 20.So d = 20 ÷ 5 = 4. The sequence increases by 4 each time.

2

Use the nth term form: T(n) = dn + c = 4n + c: We know T(3) = 13, so 4(3) + c = 13. This gives 12 + c = 13, so c = 1.

3

Write the nth term: T(n) = 4n + 1.

4

Verify with T(8): T(8) = 4(8) + 1 = 33. ✓

Final answer

T(n) = 4n + 1

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

A linear sequence has the 3rd term 13 and the 8th term 33. Find the nth term formula.

4 marks4 minslinear-sequences-worked

Show solution

Worked solution

1.Find the common difference: From term 3 to term 8 is 5 steps. The difference in values is 33 − 13 = 20.So d = 20 ÷ 5 = 4. The sequence increases by 4 each time.
2.Use the nth term form: T(n) = dn + c = 4n + c: We know T(3) = 13, so 4(3) + c = 13. This gives 12 + c = 13, so c = 1.
3.Write the nth term: T(n) = 4n + 1.
4.Verify with T(8): T(8) = 4(8) + 1 = 33. ✓

Final answer

T(n) = 4n + 1

Mark points

M1: find the common difference
M1: use the nth term form: t(n) = dn + c = 4n + c
M1: write the nth term
M1: verify with t(8)
A1: T(n) = 4n + 1

Watch out

Students confuse the common difference with the nth term itself.The common difference is the coefficient of n, not the whole formula.Also, students sometimes write the nth term using n = 1 giving T(1) = 5, then assume the formula is the first term + (n−1)×d, which is equivalent but more error-prone than the dn + c method.Always check by substituting n = 1 and n = 2.

Diagnosticrecall

Find the nth term of 3, 7, 11, 15, ...

1 mark2 minslinear-sequences-q1

Show solution

Worked solution

1.Spot the skill: The common difference d is the step between consecutive terms.
2.Use the find the common difference stage first, then use the nth term form: t(n) = dn + c = 4n + c.
3.Keep the final answer visible: 4n − 1.

Final answer

4n − 1

Mark points

M1: use the correct the common difference d is the step between consecutive terms. the nth term has the form dn + c.find d first, then substitute any known term to find c. this works even when you are not given the first term.
A1: 4n − 1

Watch out

Students confuse the common difference with the nth term itself.The common difference is the coefficient of n, not the whole formula.Also, students sometimes write the nth term using n = 1 giving T(1) = 5, then assume the formula is the first term + (n−1)×d, which is equivalent but more error-prone than the dn + c method.Always check by substituting n = 1 and n = 2.

Easyprocedure

Find the nth term of 20, 17, 14, 11, ...

2 marks3 minslinear-sequences-q2

Show solution

Worked solution

1.Spot the skill: The common difference d is the step between consecutive terms.
2.Use the use the nth term form: t(n) = dn + c = 4n + c stage first, then write the nth term.
3.Keep the final answer visible: −3n + 23.

Final answer

−3n + 23

Mark points

M1: use the correct the common difference d is the step between consecutive terms. the nth term has the form dn + c.find d first, then substitute any known term to find c. this works even when you are not given the first term.
A1: −3n + 23

Watch out

Students confuse the common difference with the nth term itself.The common difference is the coefficient of n, not the whole formula.Also, students sometimes write the nth term using n = 1 giving T(1) = 5, then assume the formula is the first term + (n−1)×d, which is equivalent but more error-prone than the dn + c method.Always check by substituting n = 1 and n = 2.

Mediumreasoning

Which term of the sequence 5n − 2 equals 98?

3 marks4 minslinear-sequences-q3

Show solution

Worked solution

1.Spot the skill: The common difference d is the step between consecutive terms.
2.Use the write the nth term stage first, then verify with t(8).
3.Keep the final answer visible: 20th term.

Final answer

20th term

Mark points

M1: use the correct the common difference d is the step between consecutive terms. the nth term has the form dn + c.find d first, then substitute any known term to find c. this works even when you are not given the first term.
A1: 20th term

Watch out

Students confuse the common difference with the nth term itself.The common difference is the coefficient of n, not the whole formula.Also, students sometimes write the nth term using n = 1 giving T(1) = 5, then assume the formula is the first term + (n−1)×d, which is equivalent but more error-prone than the dn + c method.Always check by substituting n = 1 and n = 2.

Hardproblem solving

Is 150 a term in the sequence 4n + 2? Explain.

3 marks5 minslinear-sequences-q4

Show solution

Worked solution

1.Spot the skill: The common difference d is the step between consecutive terms.
2.Use the verify with t(8) stage first, then find the common difference.
3.Keep the final answer visible: No — $14\frac{8}{4}$ = 37, so T(37) = 150 − 2 = 148, not 150.

Final answer

No — $14\frac{8}{4}$ = 37, so T(37) = 150 − 2 = 148, not 150

Mark points

M1: use the correct the common difference d is the step between consecutive terms. the nth term has the form dn + c.find d first, then substitute any known term to find c. this works even when you are not given the first term.
A1: No — $14\frac{8}{4}$ = 37, so T(37) = 150 − 2 = 148, not 150

Watch out

Students confuse the common difference with the nth term itself.The common difference is the coefficient of n, not the whole formula.Also, students sometimes write the nth term using n = 1 giving T(1) = 5, then assume the formula is the first term + (n−1)×d, which is equivalent but more error-prone than the dn + c method.Always check by substituting n = 1 and n = 2.

Exam-stylemulti-step

Two sequences are A: 3n + 1 and B: 5n − 3. Find the first term that appears in both sequences.

4 marks6 minslinear-sequences-q5

Show solution

Worked solution

1.Spot the skill: The common difference d is the step between consecutive terms.
2.Use the find the common difference stage first, then use the nth term form: t(n) = dn + c = 4n + c.
3.Keep the final answer visible: T(2) of A = 7; T(2) of B = 7; answer is 7.

Final answer

T(2) of A = 7; T(2) of B = 7; answer is 7

Mark points

M1: use the correct the common difference d is the step between consecutive terms. the nth term has the form dn + c.find d first, then substitute any known term to find c. this works even when you are not given the first term.
A1: T(2) of A = 7; T(2) of B = 7; answer is 7

Watch out

Students confuse the common difference with the nth term itself.The common difference is the coefficient of n, not the whole formula.Also, students sometimes write the nth term using n = 1 giving T(1) = 5, then assume the formula is the first term + (n−1)×d, which is equivalent but more error-prone than the dn + c method.Always check by substituting n = 1 and n = 2.

Grade 9 stretchproblem solving

The nth term of a sequence is 4n - 1. Which term has value 83?

4 marks7 minslinear-sequence-g9

Show solution

Worked solution

1.Set the nth term equal to 83.
2.Solve the equation for n.

Final answer

The 21st term

Mark points

M1: form 4n - 1 = 83
A1: n = 21

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Linear sequences - 2 marksFind the nth term of 3, 7, 11, 15, ...Mark answer

Answer

4n − 1

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

A linear sequence has a constant difference

What you will learn

Know the rule, then use it

Method

Find the common difference

Use the nth term form: T(n) = dn + c = 4n + c

Write the nth term

Watch out

A linear sequence has the 3rd term 13 and the 8th term 33. Find the nth term formula.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.