nth term of a quadratic sequence
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
Find the nth term for each sequence below.
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.