Linear Sequences for IGCSE Maths
Finding the nth term of sequences with a constant first difference. This subtopic is part of Sequences in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding li
What You Need to Know
Finding the nth term of sequences with a constant first difference. This subtopic is part of Sequences in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding linear sequences is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Linear Sequences
A linear (arithmetic) sequence has a constant difference d between consecutive terms. The nth term formula is: nth term = a + (n−1)d, which simplifies to dn + (a−d) — often written as dn + c. In IGCSE 0580, you must find the nth term formula from a sequence, find specific terms, and determine whether a given number belongs to the sequence (by solving the nth term formula = that number).
Step-by-Step Method
- 1
Find the common difference d
Subtract any term from the next: d = T₂ − T₁ = T₃ − T₂. Confirm it is constant.
- 2
Identify the first term a
The first term is a = T₁.
- 3
Write the nth term formula
nth term = a + (n−1)d = dn + (a−d). It is simpler to think: the coefficient of n is d, and the constant is T₁ − d.
- 4
Find specific terms
Substitute n = desired term number into the formula. E.g. for n = 10: T₁₀ = 10d + (a−d).
- 5
Test if a value is in the sequence
Set the nth term formula equal to the value and solve for n. If n is a positive integer, the value is in the sequence.
Worked Example
Question
A sequence begins: 7, 11, 15, 19, ... (a) Find the nth term. (b) Find the 20th term. (c) Is 97 a term in the sequence?
Solution
Part (a): d = 11 − 7 = 4. First term = 7. nth term = 4n + (7 − 4) = 4n + 3 Part (b): T₂₀ = 4(20) + 3 = 80 + 3 = 83 Part (c): 4n + 3 = 97 → 4n = 94 → n = 23.5 23.5 is not a positive integer, so 97 is NOT in the sequence. Answers: (a) 4n+3 (b) 83 (c) No
Exam Tips for Linear Sequences
- The nth term coefficient is always the common difference — not the first term.
- To find the constant c in dn + c, substitute n = 1: c = T₁ − d.
- To check if a value is in the sequence, solve for n — if n is not a whole positive number, the value is not in the sequence.
- Decreasing sequences have negative d: e.g. 20, 17, 14, 11, ... has nth term = −3n + 23.
Practice Questions
Q1: Find the nth term of: 2, 5, 8, 11, 14, ...
Show hint
d = 3. First term = 2. nth term = 3n + (2−3) = 3n − 1.
Q2: The nth term of a sequence is 5n − 2. Find the first four terms and the 100th term.
Show hint
n=1: 3, n=2: 8, n=3: 13, n=4: 18. T₁₀₀ = 498.
Q3: A sequence has nth term 7 − 3n. Is 40 a member of this sequence?
Show hint
7−3n=40 → 3n=−33 → n=−11. Negative, so not in the sequence.
Frequently Asked Questions
What is linear sequences in IGCSE Maths?
Finding the nth term of sequences with a constant first difference.
Is linear sequences in the Core or Extended syllabus?
Linear Sequences is part of the Core and Extended syllabus for IGCSE Mathematics 0580.
How do I revise linear sequences effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on linear sequences rather than trying to cover everything at once.
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