Sheet № 18 · Higher only · AQA · Edexcel · OCR
Quadratic Sequences & Nth Term –
Quadratic sequences take the pattern-spotting skills you learned with linear sequences and push them one level further. Instead of a constant first difference, quadratic sequences have a constant second difference. This topic is exclusive to the Higher tier and is tested regularly by AQA, Edexcel, and OCR. On this page you will learn how
§Key definitions
Question:
Find the nth term of 5, 12, 23, 38, 57, ...
Check:
n = 3 → 2(9) + 3 + 2 = 23 ✓; n = 5 → 2(25) + 5 + 2 = 57 ✓
Answer:
nth term = 2n² + n + 2
Q1:
Find the nth term of 4, 13, 26, 43, 64, ...
Q2:
Find the nth term of 1, 8, 19, 34, 53, ...
§Formulas to memorise
nth term = an² + bn + c
a = (second difference) ÷ 2
Write out the sequence — and label the terms T₁, T₂, T₃, etc.
Calculate the first differences — (gaps between consecutive terms).
Calculate the second differences — (gaps between the first differences). If these are constant, the sequence is quadratic.
Find a. — Divide the constant second difference by 2. This is the coefficient of n².
Subtract an² from each original term. — Write out the values of an² for n = 1, 2, 3, 4, ... and subtract them from the corresponding terms.
Find the nth term of the remaining linear sequence. — The result is bn + c.
Combine — to get the full formula: an² + bn + c.
Check — by substituting values of n back into the formula.
Worked example
Find the nth term of 5, 12, 23, 38, 57, ...
Working:
⚠ Common mistakes
- ✗Forgetting to divide the second difference by 2. The second difference is 2a, not a. If the second difference is 6, then a = 3.
- ✗Errors in the subtraction table. Calculate each value of an² carefully. A single arithmetic slip will throw off the entire linear part.
- ✗Not recognising a quadratic sequence. If the first differences are not constant, always check the second differences before assuming the sequence is something more exotic.
- ✗Mixing up first and second differences. First differences are the gaps between terms; second differences are the gaps between first differences. Label your rows clearly.
- ✗Skipping the check step. Always verify your formula against at least two original terms. This catches sign and arithmetic errors.
✦ Exam tips
- →Set out your working in a table. Examiners find this much easier to follow and it reduces your own errors. Columns for n, term, an², and the remainder work well.
- →This question is typically worth 3-4 marks. You usually get one mark for finding the second difference, one for the an² term, and one or two for the complete formula. Show each step to maximise marks.
- →Some questions give you the nth term and ask for a specific term. Simply substitute n into the formula. These are free marks if you can handle substitution into quadratics.
- →Occasionally, the sequence might start with n = 0 or use different notation. Read the question carefully to see which term corresponds to which value of n.