Quadratic Sequences for IGCSE Maths
Finding the nth term of sequences with a constant second difference. This subtopic is part of Sequences in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding quadratic s
What You Need to Know
Finding the nth term of sequences with a constant second difference. This subtopic is part of Sequences in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding quadratic sequences is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Quadratic Sequences
A quadratic sequence has a constant second difference (the differences of the differences are constant). The nth term formula contains n² and takes the form an² + bn + c. If the second difference is k, then a = k/2. IGCSE Extended tests finding the nth term of quadratic sequences and distinguishing them from linear and geometric sequences.
Step-by-Step Method
- 1
Find first and second differences
First differences: subtract consecutive terms. Second differences: subtract consecutive first differences. If second differences are constant, it is quadratic.
- 2
Find the coefficient a of n²
a = (constant second difference) ÷ 2.
- 3
Subtract the n² sequence
Subtract an² from each term to get a new linear sequence. Find its nth term (it will be bn + c).
- 4
Write the full nth term
Combine: nth term = an² + bn + c.
- 5
Verify with known terms
Substitute n = 1, 2, 3 into your formula and check they match the original sequence.
Worked Example
Question
Find the nth term of the sequence: 3, 8, 15, 24, 35, ...
Solution
First differences: 5, 7, 9, 11 (increases by 2 each time) Second differences: 2, 2, 2 — constant, so quadratic. a = 2/2 = 1, so the n² part is n². Subtract n²: n=1: 3−1=2, n=2: 8−4=4, n=3: 15−9=6, n=4: 24−16=8 This gives 2, 4, 6, 8 → nth term = 2n. Full formula: nth term = n² + 2n Verify: n=1: 1+2=3✓ n=3: 9+6=15✓ n=5: 25+10=35✓ Answer: nth term = n² + 2n
Exam Tips for Quadratic Sequences
- Always compute second differences to confirm the sequence is quadratic before applying the method.
- The coefficient of n² is HALF the second difference — forgetting to halve it is the most common error.
- After subtracting the n² part, you should always get a linear sequence — if not, recheck your second difference.
- Verify the formula with at least three terms, not just the first two.
Practice Questions
Q1: Find the nth term of: 5, 12, 21, 32, 45, ...
Show hint
Second differences = 2. a=1. Subtract n²: 4,8,12,16,20 → 4n. nth term = n² + 4n.
Q2: Find the nth term of: 3, 9, 19, 33, 51, ...
Show hint
Second differences = 4. a=2. Subtract 2n²: 1,1,1,1,1 → constant 1. nth term = 2n² + 1.
Q3: The nth term of a quadratic sequence is 2n² − n + 3. Find the 5th term and the first term greater than 100.
Show hint
T₅=50−5+3=48. Set 2n²−n+3>100 → 2n²−n−97>0. Solve.
Frequently Asked Questions
What is quadratic sequences in IGCSE Maths?
Finding the nth term of sequences with a constant second difference.
Is quadratic sequences in the Core or Extended syllabus?
Quadratic Sequences is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise quadratic 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 quadratic sequences rather than trying to cover everything at once.
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