Finding the nth Term of Linear Sequences Revision Notes | GCSE AQA Higher Further Maths

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,...

1. $8 - 5 = 3$, $\quad 11 - 8 = 3$, $\quad 14 - 11 = 3$ — The sequence increases by 3 each time, so $a = 3$. This is the coefficient of $n$. [M1]
2. $3n$ 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 $3n$.
3. $5 - 3(1) = 5 - 3 = 2$, 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]
4. $T_n = 3n + 2$ — Combine $a = 3$ and $b = 2$. [A1]
5. $T_4 = 3(4) + 2 = 12 + 2 = 14$ ✓ — 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,...

1. $17 - 20 = -3$, $\quad 14 - 17 = -3$, $\quad 11 - 14 = -3$ — The sequence decreases by 3 each time, so $a = -3$. [M1]
2. $-3n$ gives: $-3, -6, -9, -12, \ldots$ for $n = 1, 2, 3, 4$ — The actual sequence is always 23 more than $-3n$.
3. $20 - (-3)(1) = 20 + 3 = 23$, so $b = 23$ — Be careful with the sign: subtracting a negative adds. [M1]
4. $T_n = -3n + 23$ — Combine $a = -3$ and $b = 23$. [A1]
5. $T_3 = -3(3) + 23 = -9 + 23 = 14$ ✓ — 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?

1. $4n + 3 = 85$ — We are asking: for what value of $n$ does the formula give 85? [M1]
2. $4n = 82$ — Isolate the term with $n$.
3. $n = \dfrac{82}{4} = 20.5$ — $82 \div 4 = 20.5$. This is not a whole number. [A1]
4. $n = 20.5$ 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.