Nth Term of Linear Sequences –

Finding the nth term of a linear sequence is a key algebra skill at GCSE. It lets you write a rule that generates any term in a sequence without having to list them all, and it is tested on both Foundation and Higher papers across all exam boards.

What Is the Nth Term of a Linear Sequence?

A linear (arithmetic) sequence is a list of numbers where the difference between consecutive terms is always the same. This constant gap is called the common difference, d. For example, in the sequence 5, 8, 11, 14, 17, ... the common difference is 3 because each term is 3 more than the one before.

The nth term formula gives a rule for the value of any term based on its position number n. For a linear sequence, the nth term always has the form dn + c, where d is the common difference and c is a constant you calculate. Once you have the formula, you can find the 100th term, check whether a given number belongs to the sequence, or work backwards to find which term has a particular value.

The formula is sometimes written as a + (n - 1)d, where a is the first term. This is equivalent to dn + (a - d), since expanding gives dn + a - d.

Key Formulas

Step-by-Step Method

1. Find the common difference d by subtracting any term from the next term.
2. Write the "dn" part: this tells you the sequence is based on the d times table.
3. Compare the first term of the sequence with d times 1 to find the adjustment c = a - d.
4. Write the nth term as dn + c.
5. Check by substituting n = 1, 2, 3 to verify you get the original terms.

Worked Example 1 — Foundation Level

Question: Find the nth term of the sequence 4, 9, 14, 19, 24, ...

Working:

Step 1 — Common difference: 9 - 4 = 5, so d = 5.

Step 2 — The dn part is 5n.

Step 3 — When n = 1, 5n = 5, but the first term is 4. Adjustment: 4 - 5 = -1.

Step 4 — nth term = 5n - 1.

Step 5 — Check: n = 1 gives 5(1) - 1 = 4. n = 2 gives 5(2) - 1 = 9. n = 3 gives 5(3) - 1 = 14. All correct.

Answer: 5n - 1

Worked Example 2 — Higher Level

Question: The nth term of a sequence is 3n + 7. Is 52 a term in this sequence? You must show your working.

Working:

Set 3n + 7 = 52.

Subtract 7: 3n = 45.

Divide by 3: n = 15.

Since n = 15 is a positive whole number, 52 is the 15th term of the sequence.

Answer: Yes, 52 is in the sequence (it is the 15th term).

Worked Example 3 — Exam Style

Question: Here are the first five terms of a sequence: 2, -1, -4, -7, -10. Find the nth term. Find the 50th term. (3 marks)

Working:

Step 1 — Common difference: -1 - 2 = -3, so d = -3.

Step 2 — The dn part is -3n.

Step 3 — When n = 1, -3n = -3, but the first term is 2. Adjustment: 2 - (-3) = 5.

Step 4 — nth term = -3n + 5.

Step 5 — Check: n = 1 gives -3 + 5 = 2. n = 2 gives -6 + 5 = -1. Correct.

50th term: -3(50) + 5 = -150 + 5 = -145.

Answer: nth term = -3n + 5; the 50th term is -145.

Common Mistakes

• Using the wrong sign for the common difference. If the sequence decreases, d is negative. Always subtract in the correct order: next term minus current term.
• Forgetting to find the adjustment c. Writing the answer as just "5n" when the nth term is 5n - 1 loses a mark. You must include the constant.
• Confusing the first term with the zeroth term. The adjustment is found by comparing d times 1 with the first term, not d times 0.

Exam Tips

• To check if a number belongs to the sequence, set the nth term formula equal to that number and solve for n. If n is a positive integer, it is in the sequence.
• The common difference d is always the coefficient of n in the nth term formula. Use this as a quick sanity check.
• For sequences with negative common differences, be extra careful with signs when calculating the adjustment.

Practice Questions

Q1 (Foundation): Find the nth term of the sequence 7, 11, 15, 19, 23, ...

Q2 (Foundation): Find the 20th term of the sequence with nth term 6n - 5.

Q3 (Higher): The nth term of a sequence is 8n - 3. Prove that 150 is not a term in this sequence.

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Related Topics

• Sequences and Nth Term
• Quadratic Sequences
• Solving Linear Equations

Summary

• A linear sequence has a constant common difference d between consecutive terms.
• The nth term formula is dn + (a - d), where a is the first term.
• Find d by subtracting consecutive terms, then calculate the constant adjustment.
• To test whether a value belongs to the sequence, set the nth term equal to that value and check if n is a positive integer.
• Always verify your formula by substituting small values of n.