Nth term calculator for linear and quadratic sequences
Type the first few terms of your sequence and this tool works out whether it is linear or quadratic, shows the difference table, and builds the nth term rule step by step. It then lets you check any term, which is exactly how the mark scheme expects you to verify your rule.
Enter the first terms of your sequence
nth term = n² + 2n
Type: quadratic sequence
First differences: 5, 7, 9, 11
Second differences: 2, 2, 2
Term 10: 120
Working
- Second differences are all 2, so the sequence is quadratic.
- a = second difference ÷ 2 = 2 ÷ 2 = 1, giving 1n².
- Subtract 1n² from each term: 2, 4, 6, 8, 10
- The remainder is linear with difference 2, giving 2n.
- nth term = n² + 2n
The method behind it
- First differences constant → linear: nth term = dn + (first term − d).
- Second differences constant → quadratic: a = second difference ÷ 2.
- Subtract an² from each term, then find the linear rule of the remainder.
How to use this tool
1. Enter your terms
Type at least four terms separated by commas, in the order they appear in the question.
2. Read the difference table
Constant first differences mean a linear rule; constant second differences mean a quadratic rule.
3. Check a term
Enter a value of n to substitute into the rule and confirm it reproduces the sequence.
Frequently asked questions
How do you find the nth term of a linear sequence?
The common difference d is the coefficient of n. Then subtract dn from the first term to find the constant, giving nth term = dn + (first term − d).
How do you find the nth term of a quadratic sequence?
Halve the constant second difference to get the coefficient a of n². Subtract an² from each term, then find the linear nth term of what is left and add the two parts together.
How many terms do I need?
Three terms are enough for a linear sequence, but you need at least four to confirm a quadratic rule and rule out a coincidence.
What if the differences are never constant?
The sequence is not linear or quadratic. It may be geometric — each term multiplied by a fixed number — or a special sequence such as triangular or Fibonacci numbers.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.