What you'll learn
Quadratic and other non-linear sequences change by amounts that themselves change. In this guide you will learn how to recognise a quadratic sequence, how to find its nth term using second differences, how to handle other non-linear sequences (cubic, geometric, Fibonacci-type), and how to test terms. This extends the work on arithmetic sequences to higher-tier problems.
Key terms and definitions
Quadratic sequence — a sequence whose nth term contains an n² term, with a constant second difference.
First differences — the gaps between consecutive terms.
Second differences — the gaps between the first differences.
nth term — a formula for the term in position n.
Non-linear sequence — a sequence that is not arithmetic (the difference is not constant).
Core concepts
Recognising a quadratic sequence
In a quadratic sequence, the first differences change, but the second differences are constant. For example, 2, 5, 10, 17, 26: first differences 3, 5, 7, 9; second differences 2, 2, 2 — constant, so it is quadratic.
Finding the nth term
To find the nth term of a quadratic sequence:
- Find the second difference; half of it is the coefficient of n². (Second difference 2 → coefficient 1, so start with n².)
- Subtract the n² values from the sequence to leave a linear sequence.
- Find the nth term of that linear part (as for arithmetic sequences) and add it to the n² term.
For 2, 5, 10, 17, 26: coefficient of n² is 1. Subtract n² (1, 4, 9, 16, 25) to get 1, 1, 1, 1, 1 — a constant 1. So the nth term is n² + 1.
Other non-linear sequences
- Cubic sequences have a constant third difference.
- Geometric sequences multiply by a constant ratio each time.
- Fibonacci-type sequences add the previous two terms.
Identify the type by examining the differences or ratios.
Testing whether a number is in the sequence
To test a value, set the nth term equal to it and solve. For a quadratic, this gives a quadratic equation; if it has a positive whole-number solution for n, the value is in the sequence.
Worked examples
Example 1: nth term
Find the nth term of 3, 8, 15, 24, …
First differences 5, 7, 9; second difference 2, so coefficient of n² is 1. Subtract n² (1, 4, 9, 16): 2, 4, 6, 8 — linear, nth term 2n. So the nth term is n² + 2n.
Example 2: Identifying type
Is 2, 6, 18, 54 quadratic?
The terms multiply by 3 each time (constant ratio), so it is geometric, not quadratic.
Example 3: Using the nth term
For n² + 1, find the 6th term.
6² + 1 = 36 + 1 = 37.
Common mistakes and how to avoid them
Forgetting to halve the second difference. The n² coefficient is half the constant second difference.
Stopping at the n² term. Subtract n² to find the remaining linear part and add it on.
Confusing quadratic with geometric. Quadratic has constant second differences; geometric has a constant ratio.
Arithmetic slips in the differences. Compute first and second differences carefully.
Accepting non-integer n when testing. Only positive whole-number n means the value is in the sequence.
Exam technique for Quadratic Sequences
Check the second differences to confirm it's quadratic.
Halve the second difference for the n² coefficient.
Subtract the n² values and find the linear nth term.
Combine the n² and linear parts.
Identify other types (cubic, geometric, Fibonacci) by their differences or ratios.
Quick revision summary
A quadratic sequence has changing first differences but constant second differences, and its nth term contains an n² term. To find it: take the second difference and halve it for the coefficient of n² (second difference 2 → n²); subtract the n² values from the sequence to leave a linear sequence; find that linear nth term and add it to the n² term (2, 5, 10, 17, 26 → n² + 1). Other non-linear sequences include cubic (constant third difference), geometric (constant ratio), and Fibonacci-type (each term the sum of the previous two) — identify them from the differences or ratios. To test a value, set the nth term equal to it and solve, accepting only positive whole-number n. The usual errors are forgetting to halve the second difference, stopping at the n² term, confusing quadratic with geometric, and arithmetic slips. Confirm constant second differences, halve them, subtract n², find the linear part, and combine.