A sequence has a quadratic rule when its second differences are constant, and the nth term then has the form an² + bn + c. You meet it in pattern questions where the gaps grow, such as numbers of dots in a growing diagram.
This lesson belongs to the module on sequences and pattern rules and assumes you can already find a linear term rule.
Why do second differences work?
In a linear sequence the first differences are constant. In a quadratic sequence the first differences form a linear sequence of their own, which grows by a fixed amount each step. That fixed amount is the second difference.
For the rule an², the second difference is 2a. So the coefficient of n² is half the second difference. Once you know a, you subtract an² from the terms, and what remains is linear.
How to find the rule, step by step
- Write the first differences, then the second differences. Check the second differences are constant.
- Halve the second difference to get a, the coefficient of n².
- Subtract an² from each term.
- Find the linear rule bn + c for what is left, using the method from the first lesson.
- Write an² + bn + c and check with several values of n.
Worked example
Find the nth term of 4, 9, 18, 31, 48, …
Step 1, differences: first differences are 5, 9, 13, 17. Second differences are 4, 4, 4. They are constant, so the rule is quadratic.
Step 2, coefficient: half of 4 is 2, so the rule starts with 2n².
Step 3, subtract: 2n² gives 2, 8, 18, 32, 50. Then 4 − 2 = 2, 9 − 8 = 1, 18 − 18 = 0, 31 − 32 = −1, 48 − 50 = −2. The remainders are 2, 1, 0, −1, −2.
Step 4, linear part: the remainders fall by 1 each time, so the rule is −n + 3.
Step 5, rule: 2n² − n + 3.
Check: n = 1 gives 2 − 1 + 3 = 4. n = 3 gives 18 − 3 + 3 = 18. n = 5 gives 50 − 5 + 3 = 48. All match.
The mistake to watch for
The usual slip is to take the second difference as the coefficient of n².
Mistaken answer: for 3, 8, 15, 24, the second difference is 2, so the rule starts with 2n²
The student used the whole second difference.
Test it: 2n² gives 2, 8, 18, 32, and the remainders are 1, 0, −3, −8. They are not linear, so the coefficient is wrong.
Halving gives n², which leaves 2, 4, 6, 8, and so the correct rule is n² + 2n. The “subtract and look for a line” step is a built-in alarm for this error.
Check yourself
Try these without a calculator, then open each answer.
1. Find the nth term of 3, 8, 15, 24, … and then the 10th term.
Show answer
First differences: 5, 7, 9. Second differences: 2, 2. So a = 1 and the rule starts with n². Subtracting n² (1, 4, 9, 16) leaves 2, 4, 6, 8, which is 2n. The rule is n² + 2n.
10th term: 100 + 20 = 120.
2. Find the nth term of 2, 9, 20, 35, 54, …
Show answer
First differences: 7, 11, 15, 19. Second differences: 4, 4, 4. So a = 2. Subtracting 2n² (2, 8, 18, 32, 50) leaves 0, 1, 2, 3, 4, which is n − 1. The rule is 2n² + n − 1.
Check: n = 1 gives 2 + 1 − 1 = 2. n = 5 gives 50 + 5 − 1 = 54.
3. Does 1, 2, 4, 8, 16, … have constant second differences?
Show answer
First differences: 1, 2, 4, 8. Second differences: 1, 2, 4. They are not constant, so the rule is not quadratic. This sequence doubles, so it is geometric.
Where this leads next
Next, separate what n means from what a term equals in term position and term value, then use the sequences practice set. The quadratic structure explorer shows how the second difference connects to the n² term.
If you can follow a worked quadratic but get stuck on a fresh one, a teacher in online one-to-one Mathematics tuition can watch your steps and find which one slips.