Key Terms
- nth term: A formula giving any term in the sequence from its position n
- Common difference (d): The fixed amount added each term — becomes the coefficient of n
- Linear sequence: Sequence with a constant difference (nth term is dn + c)
- Position: The number of the term (1st, 2nd, 3rd…); use n in the formula
- Zero term: The value before the first term — helps find the constant c
Must-Know Facts
- The coefficient of n equals the common difference
- Find c: substitute n = 1 into dn and compare with the 1st term
- Or: find the "zero term" (term before the first) — that's c
- To find a specific term: substitute the position number for n
- To check if a value is a term: set nth term = value and check n is a positive integer
- A decreasing sequence has a negative d (e.g. 10, 7, 4, 1 → nth term = −3n + 13)
Key Formulas
- nth term = dn + c
- d = common difference (any term minus the previous term)
- c = 1st term − d
- To find the 100th term: substitute n = 100 into the formula
Common Mistakes
- c = 1st term, not d: c is found by 1st term − d, not just the first term
- Wrong difference: Always check term₂ − term₁ = term₃ − term₂ before assuming linear sequence
- Checking membership: Set dn + c = target value — if n is not a positive integer, it is not in the sequence
- Confusing position and value: The 5th term means substitute n = 5, not find term with value 5
- Negative differences: Decreasing sequences have negative d — be careful with signs when finding c
This topic summary covers Knowledge Organiser: nth Term of Linear Sequences within nth Term for GCSE Mathematics. Revise nth Term in Algebra for GCSE Mathematics with 11 exam-style questions and 4 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 8 of 8 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for nth Term
The nth term of a sequence is 3n + 2. What is the 5th term?
A student claims that 50 is a term in the sequence with nth term 3n + 1. Show whether this claim is correct.
Quick recall flashcards
What does 'n' represent?
The position number in the sequence (1st term: n=1, 2nd term: n=2, etc.)
What is the nth term?
A formula that allows you to find any term in a sequence by substituting the position number (n)