Find Any Quadratic Nth Term: A GCSE Guide

You’re probably here for one of two reasons. Either you saw a sequence in a past paper, the numbers didn’t go up by the same amount, and your brain immediately thought, “Great, now what?” Or you already know this topic appears a lot, and you want a method that works fast under pressure.

Good news. The quadratic nth term isn’t a random trick. It’s a pattern you can investigate like a detective. The numbers leave clues. Your job is to read them in the right order.

Why Mastering Quadratic Sequences Will Boost Your Grade

A lot of students lose marks on sequences because they expect every nth term question to be linear. If the terms go up by the same amount each time, that’s fine. But when the jumps between terms change, many students assume the sequence is “weird” and move on.

That’s often the exact moment the marks are hiding.

In UK GCSE maths, quadratic nth term questions appear in approximately 15 to 20% of sequence papers, and Ofqual-linked data summarised here says 78% of the 1.2 million GCSE maths entries in 2024 encountered this topic, with higher-tier papers often giving 5 to 8 marks to a question on it. That makes it a very sensible place to focus your revision energy.

For students who want structured Online Revision for GCSE, this is one of those topics worth learning properly because the same method keeps paying off.

What a quadratic sequence actually is

A quadratic sequence is a sequence where the nth term has the form:

an² + bn + c

That squared part, the n², is the giveaway. It means the pattern grows in a curved way rather than a straight-line way.

Here’s the friendly version:

• Linear sequence: add the same amount each time

• Quadratic sequence: the amount you add changes, but it changes in a steady pattern

That steady pattern is what saves you.

Practical rule: If the first differences aren’t constant, don’t panic. Check the second differences before deciding you’re stuck.

Why this feels hard at first

The topic looks intimidating because exam questions often use command words like find, derive, or verify. That can make it feel as if examiners want some magical formula pulled out of nowhere.

They don’t. They want organised thinking.

If you can spot the pattern, build a difference table neatly, and connect the clues to a, b, and c, you can turn what looks messy into a methodical puzzle. That’s exactly how strong students approach it, and it’s also how students who are trying to recover their grade can suddenly pick up marks they’d been missing.

How to Spot a Quadratic Sequence

Before you find the rule, you need to identify the type of sequence. This is the diagnosis step. If you skip it, you can end up forcing the wrong method onto the numbers.

The key clue is the second difference.

Start with the first differences

Take a sequence and subtract each term from the next one.

For example, look at this linear sequence:

3, 5, 7, 9, 11

The first differences are:

• 5 - 3 = 2

• 7 - 5 = 2

• 9 - 7 = 2

• 11 - 9 = 2

Those differences stay the same, so it’s linear.

Now compare it with this sequence:

3, 6, 11, 18, 27

The first differences are:

• 6 - 3 = 3

• 11 - 6 = 5

• 18 - 11 = 7

• 27 - 18 = 9

Those aren’t constant, so it isn’t linear.

Now check the second differences

Take the differences of the first differences:

• 5 - 3 = 2

• 7 - 5 = 2

• 9 - 7 = 2

Now the pattern is constant.

That means the original sequence is quadratic.

Sequence type	Example	First differences	Second differences
Linear	3, 5, 7, 9	2, 2, 2	0, 0
Quadratic	3, 6, 11, 18	3, 5, 7	2, 2

The detective clue to remember

Think of the first differences as footprints. If the footprints don’t match, look at how they change. If that second layer is constant, the sequence is quadratic.

Students often get confused because they expect the answer to appear in the top row straight away. It doesn’t. You have to go one level deeper.

If the first differences change by the same amount each time, you’ve found a quadratic pattern.

A neat way to lay it out

Write your sequence in rows. Don’t try to do it all in your head.

1. Write the original terms.

2. Underneath, write the first differences.

3. Under that, write the second differences.

That layout matters because later you’ll use those rows to find a, b, and c. A messy difference table causes a surprising amount of trouble in exams, not because the maths is impossible, but because one small misalignment throws off everything after it.

The Three-Step Method to Find an²+bn+c

Once you know the sequence is quadratic, the next job is to find the rule in the form:

an² + bn + c

Here, the detective work becomes very clean. You are finding three unknowns: a, b, and c.

Use this example all the way through:

2, 7, 16, 29, 46

First build the difference table.

• First differences: 5, 9, 13, 17

• Second differences: 4, 4, 4

Step 1 find a

The second difference is linked to a.

For a quadratic sequence, the second difference is always 2a.

So if the second difference is 4, then:

2a = 4

a = 2

Why does that happen? Because when you substitute whole number values into an², the squared part creates a pattern whose second difference is always double the coefficient.

You don’t need to re-prove that in the exam. You do need to trust it and use it accurately.

Step 2 find b

Now use the first term of the first differences.

For a quadratic sequence, that first difference is:

3a + b

In our example, the first difference is 5.

We already know a = 2, so:

3a + b = 5

3(2) + b = 5

6 + b = 5

b = -1

This is one of the parts students often memorise without understanding. Here’s the idea behind it. The jump from term 1 to term 2 comes from the change in the quadratic part, plus the change in the linear part. When you work that out for an² + bn + c, the first jump becomes 3a + b.

So this isn’t a random formula. It’s the first footprint left by the nth term rule.

A short video can help if you want to see the method spoken through in real time.

https://www.youtube.com/watch?v=7Isa4P8lISc

Step 3 find c

Now use the first term of the sequence itself.

When n = 1, the formula becomes:

a + b + c

The first term is 2, so:

a + b + c = 2

Substitute the values you know:

2 + (-1) + c = 2

1 + c = 2

c = 1

So the nth term is:

2n² - n + 1

Why these three clues work

Each clue comes from a different row of your table:

• Second difference gives you a

• First difference helps you find b

• First term lets you finish with c

That’s why the method feels like detective work. Each row reveals one more piece of the rule.

Check the trail: the second difference points to a, the first jump points to b, and the very first term unlocks c.

Final check

Always test your answer.

Substitute values of n:

• n = 1 gives 2(1²) - 1 + 1 = 2

• n = 2 gives 2(4) - 2 + 1 = 7

• n = 3 gives 2(9) - 3 + 1 = 16

It matches the sequence, so the rule is correct.

Worked Examples From Simple to Challenging

The method only starts to feel natural when you’ve seen it work on different types of sequence. The good news is that the logic stays the same even when the signs change.

Example 1 a straightforward positive sequence

Find the quadratic nth term of:

4, 9, 16, 25, 36

First differences:

5, 7, 9, 11

Second differences:

2, 2, 2

So:

• 2a = 2, therefore a = 1

• First difference is 5, so 3a + b = 5

• 3(1) + b = 5, so b = 2

• First term is 4, so a + b + c = 4

• 1 + 2 + c = 4, so c = 1

Nth term:

n² + 2n + 1

That can also be written as (n + 1)², but for GCSE nth term questions, n² + 2n + 1 is usually the clearest answer.

Example 2 a negative a value

Find the quadratic nth term of:

10, 13, 14, 13, 10

First differences:

3, 1, -1, -3

Second differences:

-2, -2, -2

So:

• 2a = -2, therefore a = -1

• First difference is 3, so 3a + b = 3

• 3(-1) + b = 3

• -3 + b = 3, so b = 6

• First term is 10, so a + b + c = 10

• -1 + 6 + c = 10

• 5 + c = 10, so c = 5

Nth term:

-n² + 6n + 5

Many students hesitate at this point because a negative a feels wrong. It isn’t. It means the sequence bends downward rather than upward.

Example 3 a sequence with a zero term

Find the quadratic nth term of:

0, 3, 8, 15, 24

First differences:

3, 5, 7, 9

Second differences:

2, 2, 2

So:

• 2a = 2, therefore a = 1

• First difference is 3, so 3a + b = 3

• 3(1) + b = 3, so b = 0

• First term is 0, so a + b + c = 0

• 1 + 0 + c = 0, so c = -1

Nth term:

n² - 1

That middle step is worth noticing. Sometimes b is zero. Don’t force a linear term into your answer just because you expect one.

Common Exam Pitfalls and How to Avoid Them

This topic punishes rushed setup more than difficult algebra. That’s why students can understand the idea in class but still drop marks in the exam hall.

UK examiner feedback makes that pretty clear. Edexcel GCSE Maths examiner reports summarised here noted that 28% of students lost full marks on quadratic nth term questions because they set up their equations incorrectly from the difference table, compared with 12% for linear sequences. The same source reports that only 35% of mid-tier pupils correctly identified the coefficient a without scaffolding.

Pitfall 1 misaligned differences

A very common error is writing the first differences underneath the wrong gaps.

For example, if the sequence is:

2, 7, 16, 29

the differences are:

• 7 - 2 = 5

• 16 - 7 = 9

• 29 - 16 = 13

Those numbers sit between terms, not directly under them in a casual way. If your table drifts sideways, you can end up using the wrong first difference when you solve for b.

Fix: keep your rows neat and spaced. Use squared paper if that helps.

Pitfall 2 using the wrong clue for a

Some students see a second difference of 4 and write a = 4.

That skips the key rule.

• Second difference = 2a

• So if the second difference is 4, a = 2

Exam-room habit: say quietly to yourself, “Second difference equals 2a.” That one sentence prevents a lot of lost marks.

Pitfall 3 sign mistakes with b and c

Negative numbers create the messiest slips. A student might correctly find a = -1, then make an error when solving:

3a + b = 3

They write:

3 + b = 3

instead of:

-3 + b = 3

That one sign changes the whole answer.

A similar thing happens with c when students substitute into a + b + c too quickly.

Fixes that work:

• Write brackets around negative values when substituting.

• Check one line at a time before moving on.

• Verify with substitution at the end.

If exam pressure is your bigger problem, not just maths content, it also helps to practise routines that boost your exam scores, especially techniques for slowing down just enough to avoid setup mistakes. For targeted topic practice, using real GCSE Past Papers is useful because sequence questions often reward tidy working as much as the final expression.

Quick Checks and Exam-Style Practice Questions

The fastest way to catch a wrong quadratic nth term is simple. Test the first few values.

If your answer is an² + bn + c, substitute:

• n = 1

• n = 2

• n = 3

If those don’t recreate the first three terms exactly, something has gone wrong. Usually it’s a sign error, a wrong value for a, or a misread first difference.

A quick self-check routine

Use this every time:

1. Check the second differences are constant.

2. Find a by halving the second difference.

3. Find b using the first first-difference.

4. Find c using the first term.

5. Substitute 1, 2 and 3 to verify.

A correct nth term should reproduce the original sequence, not just “look about right”.

If you want timed practice that feels more like an actual exam, Exam Practice for GCSE can help you work under pressure rather than only in calm revision mode. And if you’re exploring revision tools more broadly, it can also be useful to determine your AI expert needs so you don’t end up using something that doesn’t match the way you learn.

GCSE-Style Practice Questions

Question No.	Sequence	Difficulty
1	3, 8, 15, 24, 35	Easy
2	1, 4, 9, 16, 25	Easy
3	12, 17, 20, 21, 20	Medium
4	-2, 1, 8, 19, 34	Medium
5	15, 10, 3, -6, -17	Challenging

Try these in full exam style:

• Question 1 Find the quadratic nth term.

• Question 2 Find the quadratic nth term and write the 10th term.

• Question 3 Show that the sequence is quadratic, then find its nth term.

• Question 4 Find the nth term and verify it using the first three terms.

• Question 5 Find the nth term and state whether the sequence increases forever.

The main thing to remember is this. Quadratic nth term questions aren’t about spotting a magic formula. They’re about following clues carefully.

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