Quadratic sequences have constant second differences
What you will learn
Know the rule, then use it
These are the short notes. Read each one, then check you can use it in the worked example below.
Method
For quadratic sequences, the first differences change but the second differences are constant
Find first and second differences
First differences: 6, 10, 14, 18
Find the coefficient a
Second difference = 2a = 4, so a = 2
Subtract 2n² from the sequence to find the linear remainder
2n² gives: 2, 8, 18, 32, 50
Watch out
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused
Constant second difference means an2 + bn + c.
Multiply by the same ratio each time.
A quadratic sequence begins 3, 9, 19, 33, 51, ... Find the nth term.
Find first and second differences: First differences: 6, 10, 14, 18. Second differences: 4, 4, 4.The constant second difference confirms this is quadratic.
Find the coefficient a: Second difference = 2a = 4, so a = 2. The nth term starts with 2n².
Subtract 2n² from the sequence to find the linear remainder: 2n² gives: 2, 8, 18, 32, 50.Subtracting: 3−2=1, 9−8=1, 19−18=1, 33−32=1. The remainder is the constant sequence 1. So the linear part is 0n + 1.
Write the full nth term: T(n) = 2n² + 0n + 1 = 2n² + 1. Check: T(1) = 3 ✓, T(4) = 33 ✓.
T(n) = 2n² + 1
Build up to the hardest questions
Do them in order. If you miss a step, read the solution, then redo the question without looking.
WorkedreasoningA quadratic sequence begins 3, 9, 19, 33, 51, ... Find the nth term.
4 marks4 minsquadratic-and-geometric-sequences-workedShow solution
A quadratic sequence begins 3, 9, 19, 33, 51, ... Find the nth term.
- 1.Find first and second differences: First differences: 6, 10, 14, 18. Second differences: 4, 4, 4.The constant second difference confirms this is quadratic.
- 2.Find the coefficient a: Second difference = 2a = 4, so a = 2. The nth term starts with 2n².
- 3.Subtract 2n² from the sequence to find the linear remainder: 2n² gives: 2, 8, 18, 32, 50.Subtracting: 3−2=1, 9−8=1, 19−18=1, 33−32=1. The remainder is the constant sequence 1. So the linear part is 0n + 1.
- 4.Write the full nth term: T(n) = 2n² + 0n + 1 = 2n² + 1. Check: T(1) = 3 ✓, T(4) = 33 ✓.
T(n) = 2n² + 1
- M1: find first and second differences
- M1: find the coefficient a
- M1: subtract 2n² from the sequence to find the linear remainder
- M1: write the full nth term
- A1: T(n) = 2n² + 1
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
DiagnosticrecallFind the next two terms of the geometric sequence 5, 10, 20, 40, ...
1 mark2 minsquadratic-and-geometric-sequences-q1Show solution
Find the next two terms of the geometric sequence 5, 10, 20, 40, ...
- 1.Spot the skill: For quadratic sequences, the first differences change but the second differences are constant.
- 2.Use the find first and second differences stage first, then find the coefficient a.
- 3.Keep the final answer visible: 80, 160.
80, 160
- M1: use the correct for quadratic sequences, the first differences change but the second differences are constant.the second difference equals 2a, where t(n) = an² + bn + c.find a from the second difference, then use simultaneous equations to find b and c.
- A1: 80, 160
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
EasyprocedureFind the nth term of the geometric sequence 3, 6, 12, 24, ...
2 marks3 minsquadratic-and-geometric-sequences-q2Show solution
Find the nth term of the geometric sequence 3, 6, 12, 24, ...
- 1.Spot the skill: For quadratic sequences, the first differences change but the second differences are constant.
- 2.Use the find the coefficient a stage first, then subtract 2n² from the sequence to find the linear remainder.
- 3.Keep the final answer visible: 3 × 2^(n−1).
3 × 2^(n−1)
- M1: use the correct for quadratic sequences, the first differences change but the second differences are constant.the second difference equals 2a, where t(n) = an² + bn + c.find a from the second difference, then use simultaneous equations to find b and c.
- A1: 3 × 2^(n−1)
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
MediumreasoningFind the second differences of 1, 5, 11, 19, 29 and hence find the nth term
3 marks4 minsquadratic-and-geometric-sequences-q3Show solution
Find the second differences of 1, 5, 11, 19, 29 and hence find the nth term
- 1.Spot the skill: For quadratic sequences, the first differences change but the second differences are constant.
- 2.Use the subtract 2n² from the sequence to find the linear remainder stage first, then write the full nth term.
- 3.Keep the final answer visible: Second diff = 2, so a=1; T(n) = n² + n − 1.
Second diff = 2, so a=1; T(n) = n² + n − 1
- M1: use the correct for quadratic sequences, the first differences change but the second differences are constant.the second difference equals 2a, where t(n) = an² + bn + c.find a from the second difference, then use simultaneous equations to find b and c.
- A1: Second diff = 2, so a=1; T(n) = n² + n − 1
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
Hardproblem solvingA quadratic sequence has nth term n² − 2n + 3. Find the first term that exceeds 50.
3 marks5 minsquadratic-and-geometric-sequences-q4Show solution
A quadratic sequence has nth term n² − 2n + 3. Find the first term that exceeds 50.
- 1.Spot the skill: For quadratic sequences, the first differences change but the second differences are constant.
- 2.Use the write the full nth term stage first, then find first and second differences.
- 3.Keep the final answer visible: T(8) = 64 − 16 + 3 = 51.
T(8) = 64 − 16 + 3 = 51
- M1: use the correct for quadratic sequences, the first differences change but the second differences are constant.the second difference equals 2a, where t(n) = an² + bn + c.find a from the second difference, then use simultaneous equations to find b and c.
- A1: T(8) = 64 − 16 + 3 = 51
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
Exam-stylemulti-stepA geometric sequence has first term 4 and common ratio 0.5. Find the 5th term.
4 marks6 minsquadratic-and-geometric-sequences-q5Show solution
A geometric sequence has first term 4 and common ratio 0.5. Find the 5th term.
- 1.Spot the skill: For quadratic sequences, the first differences change but the second differences are constant.
- 2.Use the find first and second differences stage first, then find the coefficient a.
- 3.Keep the final answer visible: 0.25.
0.25
- M1: use the correct for quadratic sequences, the first differences change but the second differences are constant.the second difference equals 2a, where t(n) = an² + bn + c.find a from the second difference, then use simultaneous equations to find b and c.
- A1: 0.25
Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
Grade 9 stretchproblem solvingThe nth term of a quadratic sequence is n2 + 3n - 1. Find the first term greater than 100.
4 marks7 minssequence-g9Show solution
The nth term of a quadratic sequence is n2 + 3n - 1. Find the first term greater than 100.
- 1.Solve n2 + 3n - 1 > 100.
- 2.Test nearby positive integer values.
The 9th term, which is 107
- M1: form inequality
- M1: test n = 8 and n = 9
- A1: ninth term
Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.
Switch between skills
Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.
1Quadratic and geometric sequences - 2 marksFind the next two terms of the geometric sequence 5, 10, 20, 40, ...Mark answer
80, 160
2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
−4
3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
3x²y + xy
4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer
3x(2x + 3)
- I can explain the method for quadratic and geometric sequences.
- I can show clear working without skipping key steps.
- I can avoid this mistake: Students try to use a linear nth-term method on a quadratic sequence, finding a different step each time and getting confused.The diagnostic test is always to check second differences first. If they are constant, the sequence is quadratic.If first differences are constant, it is linear.If second differences are also changing, check for geometric (constant ratio) or other patterns.
This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.