GCSE · Maths · AQA · Spec 8300 · Foundation

Fibonacci, quadratic and geometric sequences

Some sequences add, some multiply, and some add the two terms before. You can't always tell by looking — but four quick checks will. Try them on the sort below.

Sequences · Sort by the rule

What kind of sequence is it?

Tap a sequence, then tap the family it belongs to. Test it before you choose: find the gaps between terms, then the gaps between those gaps, then try dividing.

Still to sort

Arithmetic (0)

The difference between terms is the same every time.

Where the line is: If the differences themselves change, it isn't arithmetic — look at the second differences next.

Quadratic (0)

The differences change, but the second differences (the differences of the differences) stay the same.

Where the line is: Second differences constant: quadratic. Second differences still changing: not quadratic.

Geometric (0)

You multiply by the same number each time, so each term ÷ the one before always gives the same answer.

Where the line is: Divide, don't subtract. A constant difference means arithmetic; a constant ratio means geometric.

Fibonacci-type (0)

Each term is the sum of the two terms before it.

Where the line is: The addition has to work all the way along, not just once.

None of these (0)

None of the four tests gives the same answer all the way along.

Where the line is: Only put a sequence here once all four tests have failed.

9 of 9 still to sort.

Tip: jot the gaps under the terms as you go. Most sequences give themselves away within one line of working.

Predict, then check

This time you only get three terms.

What comes next in 1, 2, 4, …?

Maths · Algebra

Using a rule you're given

Step through each line. The rule does the heavy lifting — you just follow it carefully.

GoalA Fibonacci-type sequence starts 4, b, … Its 5th term is 23. Find b.

1st = 4, 2nd = b

Give the unknown term a letter so you can build with it.

Step 1 of 8

Give the unknown term a letter so you can build with it.

Watch out: In a Fibonacci-type sequence, each new term comes from the two terms just before it — not from the first term every time.

Geometric sequences

What's the ratio really doing?

Sequence A: 48, 24, 12, 6, … Sequence B (Higher): √2, 2, 2√2, 4, …

Which of these is closest to what you think right now?

A's ratio is 24, because 48 − 24 = 24.Both are geometric: dividing each term by the one before always gives the same number.A can't be geometric. Geometric sequences always get bigger.B can't be geometric — you can't mix surds with whole numbers.

How sure are you?

I'm sureI think soI'm guessing

A's ratio is 24, because 48 − 24 = 24.

This is a common misconception

That's a difference, not a ratio — and the differences here are 24, 12, 6, so they aren't even constant. A ratio comes from dividing: 24 ÷ 48 = ½. Subtracting tests for arithmetic; dividing tests for geometric.

Both are geometric: dividing each term by the one before always gives the same number.

This matches the evidence

Exactly. A multiplies by ½ every time, and B multiplies by √2 every time. The ratio can be a whole number, a fraction or (at Higher) a surd — what makes a sequence geometric is that the ratio stays the same all the way along.

A can't be geometric. Geometric sequences always get bigger.

This is a common misconception

Only when the ratio is bigger than 1. Here 24 ÷ 48 = ½, 12 ÷ 24 = ½ and 6 ÷ 12 = ½. Multiplying by ½ halves each term, so the terms shrink — and it's still geometric. Any ratio between 0 and 1 makes the terms shrink towards zero.

B can't be geometric — you can't mix surds with whole numbers.

This is a common misconception

Divide and see: 2 ÷ √2 = √2, 2√2 ÷ 2 = √2 and 4 ÷ 2√2 = √2. The ratio is √2 every time, so B is geometric. In fact it's the powers of √2: (√2)¹, (√2)², (√2)³, (√2)⁴. Squaring √2 gives 2, which is why whole numbers keep appearing.

Divide, don't subtract. If each term ÷ the one before gives the same answer all the way along, the sequence is geometric — whether that ratio is 3, ½ or √2.

Higher

Finding the nth term of a quadratic sequence

Problem

Find an expression for the nth term of 4, 11, 22, 37, 56, …

Higher

Higher · Spot the slip

Where does this answer go wrong?

Find an expression for the nth term of 5, 14, 27, 44, 65, …

A student's answer — which line goes wrong?

First differences: 9, 13, 17, 21
Second differences: 4, 4, 4, so it's quadratic
a = 4, so the nth term starts with 4n²
4n²: 4, 16, 36, 64, 100
Terms − 4n²: 1, −2, −9, −20, −35
The leftovers drop by 3, then 7, then 11 — they aren't linear, so I'm stuck

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same difference: arithmetic. Changing differences but the same second difference: quadratic. Same ratio: geometric. Add the two before: Fibonacci-type.

What you need to know

Four tests: first differences the same → arithmetic; first differences changing but second differences the same → quadratic; each term ÷ the one before the same → geometric; each term the sum of the two before → Fibonacci-type.
The square numbers (1, 4, 9, 16, …) and the triangular numbers (1, 3, 6, 10, …) are quadratic sequences. The cube numbers (1, 8, 27, 64, …) are not.
A geometric ratio is found by dividing, never by subtracting. It can be a fraction: a ratio between 0 and 1 makes the terms shrink.
A few terms can fit more than one rule. When a sequence uses some other rule, the question gives it to you — use it forwards or backwards.
Higher: the ratio of a geometric sequence can be a surd, such as √2.
Higher: in the nth term an² + bn + c of a quadratic sequence, a is half the second difference.

The big picture

A sequence is identified by how its terms are linked, not by how it looks. Arithmetic: a constant difference. Quadratic: the differences change, but the second difference is constant — square and triangular numbers are quadratic; cube numbers are not. Geometric: a constant ratio, so you multiply by the same number r each time; the simplest are the powers of r. Fibonacci-type: each term is the sum of the two before. A few terms can fit more than one rule, so other rules are given in the question. At Higher, the ratio can be a surd, and the n² coefficient of a quadratic sequence is half its second difference.

Key points

1Arithmetic: add the same amount every time, e.g. 20, 17, 14, 11 (subtract 3 each time).

2Quadratic: the differences change by the same amount each time, e.g. 5, 8, 13, 20 (differences 3, 5, 7).

3Geometric: multiply by the same number each time, e.g. 2, 10, 50, 250 (× 5). The powers of r — r, r², r³, … — are the simplest geometric sequences.

4Fibonacci-type: add the last two terms to get the next, e.g. 2, 9, 11, 20, 31.

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Worked example

Problem

Find the next two terms of 32, 48, 72, 108, …

⚠ Watch out

Stopping a test after one or two matches. 1, 3, 4, 7, 12 looks Fibonacci-type because 1 + 3 = 4 and 3 + 4 = 7 — but 4 + 7 = 11, not 12, so it isn't. A rule only counts if it works all the way along.

🧠

Memory hook

Gap, gap of the gaps, divide, add the last two. Whichever test gives the same answer every time names the sequence.

✓

Check yourself

What type of sequence is 7, 10, 15, 22, 31, …? Write down the test that proves it, then find the next term.

Flashcards

(12)

What makes a sequence arithmetic?

The difference between consecutive terms is always the same, e.g. 5, 8, 11, 14 (+3 each time).

How do you recognise a quadratic sequence?

The first differences change, but the second differences (the differences of the differences) are constant.

Which special number sequences are quadratic?

The square numbers (1, 4, 9, 16, …) and the triangular numbers (1, 3, 6, 10, …). Both have a constant second difference.

Are the cube numbers a quadratic sequence?

No. For 1, 8, 27, 64, 125 the second differences are 12, 18, 24 — they keep changing.

How do you test whether a sequence is geometric?

Divide each term by the one before it. If the answer (the ratio) is always the same, the sequence is geometric.

What happens to a geometric sequence when its ratio is between 0 and 1?

The terms shrink towards zero, e.g. 48, 24, 12, 6 has ratio ½. It is still geometric.

What is a simple geometric progression rⁿ?

The powers of r: r¹, r², r³, … For example 3, 9, 27, 81 is 3ⁿ.

What is the rule for a Fibonacci-type sequence?

Each term is the sum of the two terms before it, e.g. 3, 4, 7, 11, 18. The Fibonacci sequence itself starts 1, 1, 2, 3, 5, 8.

Why can't three terms always tell you what a sequence is?

More than one rule can fit. 1, 2, 4 continues as 8 if you double, or as 7 if you add 1, then 2, then 3.

What do you do when a question states a rule such as 'next term = 3 × previous term − 4'?

Use exactly that rule — forwards to find later terms, or backwards (undoing each step) to find earlier ones.

Higher: what is the ratio of √2, 2, 2√2, 4, …?

√2. Each term ÷ the one before is √2, so the sequence is geometric.

Higher: how do you find a in the nth term an² + bn + c?

Halve the second difference, because the second difference of an² is 2a.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What kind of sequence is it?

Using a rule you're given

What's the ratio really doing?

Where does this answer go wrong?

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