KS3 · Maths

Geometric sequences

Some sequences add the same number every time. Geometric sequences multiply — and that changes everything.

Add, multiply — or both?

Sort the sequences

An arithmetic sequence adds the same number every time (a common difference). A geometric sequence multiplies by the same number every time — that number is the common ratio. Test each sequence two ways: subtract each term from the next, then divide each term by the one before. Tick every set it belongs to, or none.

  • A Arithmetic — the same difference every time
  • B Geometric — the same ratio every time
  1. 2, 4, 6, 8, 10
  2. 1, 2, 4, 8, 16
  3. 3, 30, 300, 3000
  4. 10 000, 10 000, 10 000, 10 000
  5. 1, 2, 6, 24
  6. 375, 75, 15, 3
  7. 2, −2, 2, −2
  8. 1, 4, 9, 16

How do you decide?

Which test would you trust?

Someone hands you the start of a sequence and asks: is it geometric?

Which is closest to how you would decide right now?
How sure are you?

Forwards and backwards

Find the missing steps

36, 48, 64 are the 2nd, 3rd and 4th terms of a geometric sequence. Find the common ratio, the 5th term and the 1st term.

  1. Divide each term by the one before: 48 ÷ 36 and 64 ÷ 48.Always test every pair, not just the first one.
  2. missing step
Which line is step 2?

What the ratio decides

One number decides what the sequence does

Choose what the common ratio is, then follow the branch to see what the sequence does.

Size of the ratio → Kind of number

5 cases to explore.

On Find the common ratio. 4 branches to choose from.

Every example here starts with a positive first term.

Lift the 5th term and watch the ratio change

123450150300450600Term numberTerm valuelift me
Common ratio 1.25

Common ratio: 1.25

Every sequence here starts at 100. Lifting the 5th term changes the common ratio, and the whole sequence follows. Above 1 the line bends upwards more and more steeply; between 0 and 1 it bends downwards; at exactly 1 it is flat. A bigger ratio makes the terms change faster.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Sequences that multiply by the same number every time — and how dividing catches them.

What you need to know

  • A geometric sequence has a first term and a common ratio: the number you multiply by to get from each term to the next.
  • Arithmetic sequences add a common difference; geometric sequences multiply by a common ratio.
  • To test for a geometric sequence, divide each term by the one before. Every answer the same → geometric.
  • Find later terms by multiplying by the ratio, and earlier terms by dividing by it.
  • With a positive first term, a ratio above 1 makes the terms grow, a ratio between 0 and 1 makes them shrink towards zero without reaching it, a ratio of 1 keeps them equal, and a negative ratio flips their sign each time.

The big picture

A geometric sequence multiplies by the same number every time; that number is the common ratio. To test a sequence, divide each term by the one before — if every answer is the same, it is geometric. Multiply by the ratio to go forwards and divide by it to go backwards. The size of the ratio decides whether the terms grow, shrink, stay the same or flip between positive and negative.

Key points

1Geometric = multiply by the same number every time.
2Common ratio = any term ÷ the term before it.
3Forwards: × the ratio. Backwards: ÷ the ratio.
4Keep non-whole ratios as fractions, like 4/3 — not 1.33.
5A constant sequence is both arithmetic (difference 0) and geometric (ratio 1).
6Plotted terms curve upwards for a ratio above 1 and downwards for a ratio between 0 and 1 — but only the whole-number positions are terms.

Worked example

Problem

The sequence 48, 12, 3, … is geometric. Find the common ratio and the next two terms.

⚠ Watch out

Deciding from the first two terms. 3, 6 could continue 9 (adding 3) or 12 (multiplying by 2) — divide every pair of neighbours before you decide.

🧠

Memory hook

Adding? Subtract to check. Multiplying? Divide to check.

✓

Check yourself

A geometric sequence has a common ratio of 1/2. Explain why its terms can never reach zero, however many you write down.

Flashcards

(15)
What is a geometric sequence?
A sequence where you multiply by the same number to get from each term to the next.
What is the common ratio of a geometric sequence?
The fixed multiplier that links every term to the next.
How do you test whether a sequence is geometric?
Divide each term by the one before. If every answer is the same, it is geometric.
Arithmetic or geometric: which adds and which multiplies?
Arithmetic adds a common difference. Geometric multiplies by a common ratio.
If the multiplier changes from term to term, is the sequence geometric?
No. A geometric sequence multiplies by the same number every time.
Why can't the first two terms tell you what type a sequence is?
Any two terms can be linked by adding or by multiplying. You need to check every pair of neighbours.
Moving along a geometric sequence: what do you do going forwards, and going backwards?
Forwards, × the ratio. Backwards, use the inverse: ÷ the ratio.
Why write a common ratio of 4/3 as a fraction rather than 1.33?
4/3 is exact; 1.33 is rounded, and the error grows with every term you work out from it.
What does a common ratio between 0 and 1 do (positive first term)?
The terms get smaller each time but never reach zero.
Does a fraction as the common ratio always make a sequence shrink?
No. A fraction bigger than 1, such as 5/4, makes it grow.
What does a negative common ratio do?
The terms switch between positive and negative.
What does a common ratio of 1 give?
Every term is the same.
Can a sequence be both arithmetic and geometric?
Yes — a constant sequence, with a common difference of 0 and a common ratio of 1.
What shape do the plotted terms make when the ratio is greater than 1?
An upward curve that gets steeper and steeper.
Why doesn't a geometric sequence have a value at position 2½?
Its terms are discrete — only whole-number positions are terms. A line through the points only shows the shape.

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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