Teaching unit · UK Year 8 (Key Stage 3, ages 12 to 13)

Geometric sequences

Recognising, continuing and finding the nth term of a sequence that grows or shrinks by a constant ratio

About two lessons of 45 to 60 minutes

Start here · hook

Add the same amount every time, or MULTIPLY by the same amount every time

An arithmetic sequence, like 3, 7, 11, 15, grows by ADDING the same fixed amount each time (here, +4). A GEOMETRIC sequence grows in a completely different way, by MULTIPLYING by the same fixed amount, called the COMMON RATIO, each time. A bacteria colony that doubles every hour (1, 2, 4, 8, 16...) is geometric. So is money that halves in value every year through depreciation, or a bouncing ball that reaches half the height of its previous bounce.

The 2 patterns can look deceptively similar at a glance (both are 'sequences that grow'), but they behave very differently: an arithmetic sequence eventually gets left far behind by a geometric one growing from the same starting point, because multiplying compounds far faster than adding.

A bacteria colony doubling every hour: 1, 2, 4, 8, 16common ratio 2 (multiply by 2 each hour)
A ball bouncing to half its previous height: 160cm, 80cm, 40cm, 20cmcommon ratio 1/2 (halving each bounce)
Savings tripling every year: £5, £15, £45, £135common ratio 3
Folding paper in half repeatedly: 1, 2, 4, 8, 16 layersthe same doubling pattern as the bacteria example

Learning objective

What students will be able to do

Students will recognise a geometric sequence by testing for a constant common ratio (dividing consecutive terms), distinguish geometric from arithmetic sequences, continue a geometric sequence, and write and evaluate its nth-term formula a x $r^{n-1}$ .

Success criteria

I can test whether a sequence is geometric by checking that dividing consecutive terms always gives the same common ratio.
I can find the next terms of a geometric sequence, given the first few terms.
I can explain the difference between an arithmetic sequence (constant difference) and a geometric sequence (constant ratio).
I can write the nth-term formula of a geometric sequence as a x $r^{n-1}$ , and evaluate it for a given n.

Curriculum anchor

Standards this unit teaches

KS3 Maths: AlgebraUK National Curriculum (England)
Algebra
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Algebra" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "generate terms of a sequence from either a term-to-term or a position-to-term rule"; "recognise arithmetic sequences and find the nth term" (arithmetic sequences; the UK KS3 batch 3 push, lib/content_uks3math3.ts, already built this half); "recognise geometric sequences and appreciate other sequences that arise" (the geometric half, built here).

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Geometric sequence: a sequence where each term is found by multiplying the previous term by a fixed number (the common ratio)
Common ratio: the fixed number a geometric sequence is multiplied by from 1 term to the next, often written r
Arithmetic sequence: a sequence where each term is found by adding a fixed number (the common difference) to the previous term
nth term: a formula that gives any term of a sequence directly, without listing every term before it
Exponential growth: the rapid style of growth produced by repeated multiplication, the pattern behind every geometric sequence

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Recognising a geometric sequence

Concrete

To test whether a sequence is geometric, divide each term by the term before it. If that division gives the SAME answer every time, the sequence is geometric, and that constant answer is the common ratio. If instead SUBTRACTING consecutive terms gives the same answer every time, the sequence is arithmetic instead.

The geometric sequence 1, 2, 4, 8, 16: each term is double the one before it, so it does NOT sit at evenly-spaced points, unlike an arithmetic sequence with a constant difference.

Worked example

Is 3, 6, 12, 24 arithmetic or geometric?

Check the differences: 6 - 3 = 3, 12 - 6 = 6. Not constant, so not arithmetic.
Check the ratios: 6 / 3 = 2, 12 / 6 = 2, 24 / 12 = 2. Constant, so it is geometric with common ratio 2.

Answer: Geometric, common ratio 2.

Check for understanding, ask

Why is dividing (not subtracting) the right test for a geometric sequence?
Is 5, 8, 11, 14 arithmetic or geometric? Explain.

2. Continuing a sequence, including shrinking sequences

Pictorial

A common ratio bigger than 1 makes a sequence GROW (like 1, 2, 4, 8). A common ratio between 0 and 1 (like 1/2) makes a sequence SHRINK toward 0, dividing each term to get the next. Both are still geometric sequences, since the underlying test (a constant ratio between consecutive terms) still holds.

Worked example

Find the next term of 162, 54, 18, 6, and state the common ratio.

Check the ratio: 54 / 162 = 1/3, 18 / 54 = 1/3, 6 / 18 = 1/3. The common ratio is 1/3.
Next term: 6 x 1/3 = 2 (equivalently, 6 / 3 = 2).

Answer: Next term: 2. Common ratio: 1/3.

Check for understanding, ask

A sequence has common ratio 1/2. Will it ever reach exactly 0? Explain.

3. The nth-term formula

Abstract

Just as an arithmetic sequence has the nth-term formula a + (n - 1)d, a geometric sequence has the nth-term formula a x $r^{n-1}$ , where a is the first term and r is the common ratio. This lets you jump straight to any term (like the 20th) without listing every term before it.

Worked example

A geometric sequence has first term 3 and common ratio 2. Find the 6th term.

nth term = a x $r^{n-1}$ = 3 x $2^{n-1}$ .
For n = 6: 3 x $2^{5}$ = 3 x 32 = 96.

Answer: 96.

Check for understanding, ask

Why is the exponent (n - 1) and not n in the nth-term formula?
A geometric sequence has first term 2 and common ratio 3. Find the 4th term using the formula.

Watch for

Common misconceptions and how to address them

MisconceptionYou can test for a geometric sequence by subtracting consecutive terms, the same way you test for arithmetic.

Why it happens: Students default to the more familiar 'subtract to check' test learned for arithmetic sequences, without realising a different sequence TYPE needs a different test.

How to address it: Always try BOTH tests explicitly (subtract, then divide) side by side, and label which one gives a constant result, making the choice of test (not just the answer) visible.

MisconceptionThe nth term of a geometric sequence with first term a and ratio r is a x $r^{n}$ .

Why it happens: The exponent pattern is over-generalised from the term NUMBER rather than the number of MULTIPLICATIONS actually applied (the 1st term needs 0 multiplications, the 2nd term needs 1, and so on).

How to address it: List the exponents explicitly next to each term: term 1 = a x $r^{0}$ , term 2 = a x $r^{1}$ , term 3 = a x $r^{2}$ , making the pattern 'exponent is always 1 less than the term number' visible.

MisconceptionA common ratio less than 1 means the sequence is not really geometric, since it 'goes backward'.

Why it happens: Confusing the word 'ratio' with an expectation of growth, when a ratio can validly be less than 1 (still a constant multiplier, just a shrinking one).

How to address it: Show that the SAME test (divide consecutive terms, get a constant) succeeds for a shrinking sequence like 100, 50, 25, exactly as it does for a growing one.

Do it together

Guided practice (with answers)

1. Is 2, 6, 18, 54 arithmetic or geometric? State the common ratio or common difference.
Answer: Geometric, common ratio 3, because 6/2 = 18/6 = 54/18 = 3.
2. Is 4, 9, 14, 19 arithmetic or geometric? State the common ratio or common difference.
Answer: Arithmetic, common difference 5, because 9-4 = 14-9 = 19-14 = 5.
3. Find the next 2 terms of 5, 10, 20, 40.
Answer: 80, 160, because the common ratio is 2.
4. Find the next term of 200, 100, 50, 25.
Answer: 12.5, because the common ratio is 1/2.
5. A geometric sequence has first term 1 and common ratio 4. Write the nth-term formula and find the 5th term.
Answer: nth term = 1 x $4^{n-1}$ . 5th term = $4^{4}$ = 256.
6. A geometric sequence has first term 5 and common ratio 2. Find the 4th term using the nth-term formula.
Answer: 5 x $2^{3}$ = 5 x 8 = 40.

On their own

Independent practice worksheets

Practise recognising, continuing and finding the nth term of geometric sequences with computed, never-wrong answer keys.

Year 8 geometric sequences worksheetall 4 skills from this unit, freshly generated each time

Reach every student

Differentiation

Support

Only use whole-number growing ratios (2, 3, 4) until the concept is secure before introducing shrinking (fractional) ratios.
Write each step of a sequence as a repeated multiplication chain (a, a x r, a x r x r, ...) before compressing it to the $r^{n-1}$ formula.
Provide a side-by-side arithmetic vs geometric example pair for every new sequence, so the distinguishing test is always freshly practised.

Extension

Investigate what happens to a geometric sequence when the common ratio is negative (the terms alternate sign).
Explore why a geometric sequence with 0 < r < 1 gets closer and closer to 0 but (in this KS3 context) never reaches it exactly.
Find the common ratio and the first term, given only the 3rd and 5th terms of a geometric sequence.
Compare the long-run size of an arithmetic and a geometric sequence that start at the same value, and explain why the geometric one eventually overtakes.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling recognition, continuation and the nth-term formula.

1. Is 7, 14, 28, 56 arithmetic or geometric? State the common ratio or difference.
Answer: Geometric, common ratio 2.
2. Find the next term of 81, 27, 9, 3.
Answer: 1, because the common ratio is 1/3.
3. A geometric sequence has first term 2 and common ratio 3. Find the 5th term.
Answer: 2 x $3^{4}$ = 2 x 81 = 162.

For the teacher

Teacher notes and timings

Rough timing across 2 lessons: Lesson 1 recognising and continuing sequences (sections 1-2), Lesson 2 the nth-term formula (section 3) plus the exit ticket.
This unit deliberately covers ONLY the geometric half of the KS3 sequences bullet; arithmetic sequences and their nth term (a + (n-1)d) were already fully built in the UK KS3 batch 3 push (lib/content_uks3math3.ts, lib/teachingUnits_secondary.ts's earlier sequences unit), so this unit does not duplicate that content, only contrasts against it.
Every shrinking-sequence example in the matching worksheet uses a first term constructed as a multiple of the ratio's denominator raised to a high enough power (e.g. 96, 48, 24... for ratio 1/2), so every term stays a whole number, never a rounded decimal.

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