GCSE Maths Quadratic Sequences and Nth Terms
Quadratic sequences have a constant second difference, even though their first differences change. GCSE Higher questions ask you to identify an nth-term rule such as n² + 1 or 2n² − 3n + 4. The safest method is to inspect the differences, find the coefficient of n² and then compare the sequence with the corresponding square-number pattern.
Recognise a constant second difference
Consider 2, 5, 10, 17, 26. The first differences are 3, 5, 7 and 9. The second differences are 2, 2 and 2, so the sequence is quadratic. An arithmetic sequence would have constant first differences. Write a simple difference table beneath the original values rather than trying to guess the formula immediately. Consistent second differences are the clue that the nth term contains an n² component.
Further reading: Year 8 Maths Sequences and Nth Terms
Find the coefficient of the square term
For a quadratic rule an² + bn + c, the constant second difference is 2a. Therefore, when the second difference is 2, a = 1. Begin with n², whose first five values are 1, 4, 9, 16 and 25. Compare these with 2, 5, 10, 17 and 26. Each actual term is exactly one higher, giving the nth-term rule n² + 1. Test n = 5 to check 25 + 1 = 26.
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A pattern with a linear adjustment
Take 3, 8, 15, 24. The first differences are 5, 7, 9 and the second difference is 2, so begin with n². Subtract 1, 4, 9, 16 from the original terms and the remainders are 2, 4, 6, 8. These form the linear pattern 2n, giving the nth term n² + 2n. Verify the third term: 3² + 2 × 3 = 9 + 6 = 15.
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When the second difference is greater than two
If the second difference is 6, the coefficient of n² must be 3. Begin with the pattern 3n²: 3, 12, 27, 48 and so on. Subtract these from the target sequence and inspect the remainder. If the remainder changes by a constant amount, represent it as bn + c. This staged method is generally easier to check than immediately forming simultaneous equations from the first three terms.
Further reading: GCSE Maths Completing the Square Step by Step
Find a distant term efficiently
For the rule n² + 2n, the tenth term is 10² + 2(10) = 120. There is no need to write the next six sequence values one by one. Explain that n is the positive integer position of a term and not the term’s numerical value. A formula should also reproduce the first terms; checking a distant term alone does not establish that your guessed rule is correct.
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Avoid the most common mistakes
Students sometimes confuse first and second differences, use the second difference itself as the coefficient of n², or forget the linear and constant terms. Divide the second difference by two to obtain a, then compare the actual sequence with an². Keep subtraction careful when negative adjustments occur. Finally, substitute at least three term positions to verify the complete rule.
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Frequently asked questions
Is every sequence with increasing gaps quadratic?
No. Constant second differences are the characteristic test for a quadratic sequence.
Why divide the second difference by two?
The second differences of an² are constant and equal to 2a.
Related GCSE revision resources
The existing guides below explore supporting skills. Choose the explanations that match the area where further understanding is required.
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