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# nth term of a quadratic sequence
- URL: https://www.esheets.io/nth-term-of-a-quadratic-sequence/
- Published: 2025-09-25T15:21:26.000Z
- Updated: 2026-07-09T20:31:08.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

Finding the **nth term** of a sequence helps you predict any number in a pattern without listing them all out. Whether you're working out how many seats are in each cinema row or tracking how a saving plan grows week by week, the nth term gives you the formula behind the pattern. [Jump to the questions](#practise-now)

[Looking for questions on generating quadratic sequences?](https://www.esheets.io/generating-quadratic-sequences/)

## Practise now

Find the nth term for each sequence below.

Worksheet preview and key skills 

### Worksheet preview

Practise the nth term of a quadratic sequence with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Finding first and second differences.
- Using the second difference to identify the n² term.
- Adjusting the expression to match the sequence.
- Writing the quadratic nth term rule.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Topic guide

### What this worksheet practises

This worksheet provides practice on finding the exact "nth term" algebraic formula for a quadratic sequence (e.g. finding that the rule is 2n² + 3n − 1). This is a multi-step algebraic process and is significantly harder than finding linear nth terms.

### Key method

You must find the first and second differences of the sequence to begin.

- **Step 1 (The a value):** Calculate the gaps between the numbers (the first differences). Then calculate the gaps between those gaps (the second differences). The second difference will be a constant number. **Halve this constant number.** This gives you 'a', the number in front of your n².
- **Step 2 (The linear sequence):** Write out the sequence generated by just your an² term (substitute n=1, n=2, n=3).
- **Step 3:** Subtract your an² sequence from the original sequence given in the question.
- **Step 4:** The result of this subtraction will be a normal linear sequence. Find the nth term of this new linear sequence.
- **Step 5:** Combine your 'an²' part and your linear part to create the final full formula.

### Worked example

**Find the nth term of: 5, 12, 23, 38...**

Step 1: First differences are 7, 11, 15\. The second difference is 4\. Halve it. Our first term is **2n²**.

Step 2: Generate the 2n² sequence (2×1², 2×2², 2×3²): 2, 8, 18, 32.

Step 3: Subtract this from the original sequence.

Original: 5, 12, 23, 38  
Subtract: 2, 8, 18, 32  
Result: 3, 4, 5, 6.

Step 4: Find the nth term of the result (3, 4, 5, 6). It goes up by 1, and the 'zeroth' term is 2\. So the linear rule is **1n + 2**.

Step 5: Combine them. The final formula is **2n² + n + 2**.

### Common mistakes to avoid

The single most common mistake is forgetting to *halve* the second difference in step 1\. If the second difference is 4, students often write 4n² instead of 2n². This completely derails the rest of the calculation.

### How to check your answer

Always test your final formula using a number further down the sequence. For example, test n=3\. Using our final formula: 2(3)² + 3 + 2 = 2(9) + 5 = 18 + 5 = 23\. This matches the third number in the original sequence, proving the formula is correct.