Finding the nth Term of Linear Sequences Revision Notes | GCSE AQA Higher Further Maths
The common difference becomes the coefficient of n, then subtract it from the first term to find the constant. Notes, worked examples and 8 questions answered.
Updated 14 Mar 2026 · Revise Right Team
Finding the nth Term of Linear Sequences is part of nth terms of linear sequences in GCSE AQA Higher Further Maths.
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What this lesson covers
The common difference becomes the coefficient of n, then subtract it from the first term to find the constant. Notes, worked examples and 8 questions answered.
Find the nth term formula for a linear sequence by identifying the common difference and first term
Apply the nth term formula to calculate any term in a linear sequence
Verify whether a given value is a term in a linear sequence using the nth term formula
Identify the common difference and its role as the coefficient of n in the nth term formula
Key ideas to keep in view
Use this lesson to stay inside nth terms of linear sequences while connecting the detail back to the wider Algebra topic.
Linear sequence: A sequence in which consecutive terms differ by a constant amount. Used when a pattern increases or decreases by the same value each time.
Common difference: The fixed amount added (or subtracted) between consecutive terms in a linear sequence. It becomes the coefficient of n in the nth term formula.
nth term formula: An algebraic expression of the form $an + b$ that gives the value of any term in a linear sequence when n is substituted. Used to find any term without listing the whole sequence.
Revision notes
Read the core explanations from Finding the nth Term of Linear Sequences before testing yourself from memory.
1. What Is a Linear Sequence and Why Does It Matter?
Imagine you are saving £5 every week. After week 1 you have £5, after week 2 you have £10, after week 3 you have £15. That pattern — growing by the same amount each time — is a linear sequence. The question examiners love to ask is: how much will you have after week 47? Listing 47 terms is painful. A formula makes it instant.
Finding the nth term of a linear sequence is one of the most reliably tested skills at GCSE. Master the method once and you can answer any variation the examiner sets.
2. The Structure of a Linear Sequence
Every linear sequence has a common difference, and that difference is the engine of the nth term formula. Here is the key insight: the common difference becomes the coefficient of $n$ in the formula $an + b$.
The graph makes the link concrete: a linear sequence plots as a straight line because the difference between consecutive terms is constant. The gradient of that line is exactly the common difference $d$.
3. The nth Term Formula
The method has two clear stages: find $a$ (the common difference), then find $b$ (the adjustment). Once you have both, you can find any term or check whether a number belongs to the sequence.
3a. Step-by-Step Method
Worked example: Finding the nth Term: Sequence 5, 8, 11, 14,...
Find the nth term of the sequence 5, 8, 11, 14,...
$8 - 5 = 3$, $\quad 11 - 8 = 3$, $\quad 14 - 11 = 3$ — The sequence increases by 3 each time, so $a = 3$. This is the coefficient of $n$. [M1]
$3n$ gives: $3, 6, 9, 12, \ldots$ for $n = 1, 2, 3, 4$ — Compare $3n$ with the actual sequence. The actual sequence is always 2 more than $3n$.
$5 - 3(1) = 5 - 3 = 2$, so $b = 2$ — Subtract $a \times 1$ from the first term to find $b$. You can check with any term: $8 - 3(2) = 2$. [M1]
$T_4 = 3(4) + 2 = 12 + 2 = 14$ ✓ — The 4th term of the original sequence is 14, which matches. Always check at least one term. [A1]
Answer: $T_n = 3n + 2$
4. A Worked Example with a Decreasing Sequence
Decreasing sequences catch students out because the common difference is negative. The method is identical — you need to handle the negative value carefully when finding $b$.
Worked example: Finding the nth Term: Sequence 20, 17, 14, 11,...
Find the nth term of the sequence 20, 17, 14, 11,...
$17 - 20 = -3$, $\quad 14 - 17 = -3$, $\quad 11 - 14 = -3$ — The sequence decreases by 3 each time, so $a = -3$. [M1]
$-3n$ gives: $-3, -6, -9, -12, \ldots$ for $n = 1, 2, 3, 4$ — The actual sequence is always 23 more than $-3n$.
$20 - (-3)(1) = 20 + 3 = 23$, so $b = 23$ — Be careful with the sign: subtracting a negative adds. [M1]
$T_3 = -3(3) + 23 = -9 + 23 = 14$ ✓ — The 3rd term of the original sequence is 14. The formula is correct.
Answer: $T_n = -3n + 23$
5. Using the nth Term to Find a Specific Term or Check Membership
Once you have the formula, two further question types become straightforward. The first asks you to calculate a specific term — substitute the term number and evaluate. The second asks whether a given value is a term in the sequence — set the formula equal to that value and solve for $n$.
The membership check is where many students lose marks. If $n$ works out to a positive whole number, the value is in the sequence. If $n$ is a fraction or a negative number, it is not.
Worked example: Is 85 a Term in the Sequence with nth Term $4n + 3$?
The nth term of a sequence is $4n + 3$. Is 85 a term in this sequence?
$4n + 3 = 85$ — We are asking: for what value of $n$ does the formula give 85? [M1]
$4n = 82$ — Isolate the term with $n$.
$n = \dfrac{82}{4} = 20.5$ — $82 \div 4 = 20.5$. This is not a whole number. [A1]
$n = 20.5$ is not a positive integer — Term numbers must be positive whole numbers. Since $n = 20.5$, the value 85 is NOT a term in this sequence.
Answer: 85 is not a term in the sequence because $n = 20.5$ is not a whole number.
7. Deliberate Practice
These questions cover the full range of what the examiner can ask. Work through each one before checking the answer.
8. Exam Technique and Common Mistakes
The examiner has a small set of traps they return to repeatedly. Knowing them in advance turns potential dropped marks into guaranteed ones.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.
How do you find the nth term of a linear sequence?
The nth term of a linear sequence is found by working out the common difference, which becomes the coefficient of n, then finding the constant by subtracting it from the first term. For 5, 8, 11, 14, the difference is 3, so start with 3n; then 5 − 3 = 2, giving Tₙ = 3n + 2, and T₄ = 14 checks.
What is the common difference of a sequence?
The common difference d is the value added to each term to get the next. If a sequence goes up by 3 each time, d = 3; if it goes down by 4, d = −4. In a linear sequence, the common difference is the coefficient of n in the nth term an + b.
Why does a linear sequence make a straight-line graph?
A linear sequence plots as a straight line because the difference between consecutive terms is constant. The gradient of that line is exactly the common difference d, which is also the coefficient of n in the nth term formula.
How do you find the nth term of a decreasing sequence?
The nth term of a decreasing sequence uses the same method, but the common difference is negative. For 20, 17, 14, 11, the difference is −3, so start with −3n; then 20 − (−3)(1) = 23, giving Tₙ = −3n + 23, and T₃ = −9 + 23 = 14 checks.
How do you check if a number is in a sequence?
Set the nth term formula equal to the number and solve for n. If n is a positive whole number, the value is in the sequence. For 4n + 3 = 85, 4n = 82 and n = 20.5, which is not a whole number, so 85 is not a term.
Why is the common difference the coefficient of n?
The common difference is the coefficient of n because each time n increases by 1, the term increases by that amount. A linear sequence plots as a straight line, and the gradient of that line is exactly the common difference d.
What does it mean if n is not a whole number?
If solving for n gives a fraction or a negative number, the value is not a term in the sequence, because term numbers must be positive whole numbers. For 4n + 3 = 85, n = 82 ÷ 4 = 20.5, so 85 is not a term.
How do you find a specific term using the nth term?
A specific term is found by substituting its term number into the nth term formula and evaluating. For Tₙ = 3n + 2, the 4th term is 3(4) + 2 = 14, and any other term can be found the same way without listing the sequence.