Key Stage 4 maths · Algebra

GCSE higher: the nth term of quadratic sequences explained

This is the higher-tier extension of sequences. Pupils find the nth term of a quadratic sequence by using first and second differences, work with geometric progressions whose common ratio is a surd such as √2, and meet other sequences defined by a given rule. It combines algebra, number patterns and surd arithmetic, so it rewards careful, organised working.

Quadratic sequences and surd ratios, explained

In a quadratic sequence the first differences are not constant, but the second differences, the differences between the differences, are. The nth term has the form an² + bn + c. Take 5, 12, 23, 38, 57. The first differences are 7, 11, 15, 19 and the second differences are 4, 4, 4. A constant second difference is the signal that the nth term contains n² and no higher power.

The method is the same every time. Find the second difference and halve it to get a, the number in front of n². Subtract an² from each term of the sequence. What remains is a linear sequence, whose nth term is found as in Key Stage 3 and earlier GCSE work. Add the two parts together. The halving is important: the sequence of square numbers 1, 4, 9, 16 has a second difference of 2, so the second difference of any quadratic sequence is always twice a. If the second differences are negative, a is negative and the sequence eventually falls.

Once pupils have the nth term, they can use it to check whether a number is in the sequence. For 2n² + n + 2, setting 2n² + n + 2 = 107 gives 2n² + n − 105 = 0, which factorises as (2n + 15)(n − 7) = 0, so n = 7 and 107 is the 7th term. The other solution is negative and cannot be a position. For 100, the equation 2n² + n − 98 = 0 has no whole-number solution, so 100 is not in the sequence.

A geometric progression multiplies by the same number each time. At this level the common ratio can be a surd. For 1, √2, 2, 2√2, 4 the ratio is √2, because √2 × √2 = 2. Each term is double the term two places before it. Pupils find the ratio by dividing a term by the one before, and can work with missing terms: if the first term is 4 and the third is 12, then r² = 3, so r = √3 (taking the positive root) and the second term is 4√3. This draws on the rules for multiplying and simplifying surds.

The programme of study also says pupils should recognise and use “other sequences” without listing them. In practice that means being ready to work from any rule they are given: a term-to-term rule that uses the previous term, such as “double it and add 1”, or an nth-term rule with a power in it, such as n³ + 1. Specifications differ in what they list, so pupils should be confident generating terms by substitution and looking at differences to decide what kind of sequence they have.

Words your child will hear

First difference: The result of subtracting each term from the next. 5, 12, 23 has first differences 7 and 11.
Second difference: The differences between the first differences. If these are constant, the sequence is quadratic.
Quadratic sequence: A sequence with a constant second difference, whose nth term has an n² part: an² + bn + c.
Common ratio: The number multiplied each time in a geometric progression, found by dividing a term by the one before.
Surd: A root that is not a whole number or fraction, such as √2 or √5, left in root form so the answer stays exact.
Term-to-term rule: A rule that gives each term from the previous one, like “double and add 1”.
Coefficient: The number in front of a letter. In 2n² + n + 2, the coefficient of n² is 2.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Recognise and use simple geometric progressions where the common ratio r is a surd, and other sequences
Deduce expressions to calculate the nth term of quadratic sequences

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Find the nth term of the sequence 5, 12, 23, 38, 57.

Find the first differences: 12 − 5 = 7, 23 − 12 = 11, 38 − 23 = 15, 57 − 38 = 19. They are not constant, so the sequence is not linear.
Find the second differences: 11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4. They are constant, so the sequence is quadratic. Half of 4 is 2, so the nth term starts with 2n².

Subtract 2n² from each term, using the squares 1, 4, 9, 16, 25 doubled.

n          1    2    3    4    5
2n²        2    8   18   32   50
term       5   12   23   38   57
term - 2n² 3    4    5    6    7

What is left, 3, 4, 5, 6, 7, is linear with difference 1, so it is n + 2.
Put the parts together: 2n² + n + 2. Check with n = 4: 2 × 16 + 4 + 2 = 38, which matches the sequence.

The nth term is 2n² + n + 2.

Common mistakes and how to help

Using the second difference as the number in front of n²

The second difference is 4, so pupils write 4n². But 4n² would give a second difference of 8.

What helps: Remember that n² on its own has second difference 2, so a is always half of it. Test the 1st and 2nd terms of your answer straight away.

Trying the linear method on the original terms

Pupils are used to finding the difference and writing a number times n. With a quadratic sequence the differences keep changing, so that approach gives a wrong rule.

What helps: Check the first differences first. If they are not constant, look at the second differences, then subtract an² before looking for the linear part.

Not checking the finished nth term

The method has several short steps, and a slip in any of them gives an answer that looks plausible.

What helps: Substitute n = 1, 2 and 3 into the answer, then try a term you did not use in the working, such as n = 5. If all of them match, the rule is right.

Getting surd multiplication wrong when finding terms

√2 × √2 is often written as 2√2, which mixes up adding and multiplying surds. √2 + √2 = 2√2 but √2 × √2 = 2.

What helps: Think of a square of side √2, which has area 2. Whenever a surd is multiplied by itself, the root disappears.

How to help at home

Make a sequence from a rule. Choose an nth term such as 3n² − 1 and write the first four terms: 2, 11, 26, 47. Give them to your teenager to find the rule, then swap roles. Setting questions is good revision.
Grow square patterns with counters. Make a 1 by 1, 2 by 2, 3 by 3 and 4 by 4 square from counters. The number added each time is 3, then 5, then 7. Talk about why the second difference is always 2.
Draw √2 on squared paper. Draw a diagonal of a 1 by 1 square. That length is √2. Then draw a square on that diagonal and count its area in half squares: it is 2, which shows √2 × √2 = 2.
Spot the odd one out. Write a sequence like 2, 5, 10, 17, 26 and a lookalike with one wrong term. Ask your teenager to find the odd one using first and second differences.

Builds on

Key Stage 3: Algebraic notation and formulae
Key Stage 3: Solving linear equations
Key Stage 3: Graphs of linear and quadratic functions
Key Stage 3: Sequences and the nth term