What even is this? A geometric sequence multiplies by the same number at every step, unlike arithmetic sequences, which add. Doubling every hour? Halving each year? Bacteria? Investments? Radioactive decay? All geometric. The exam loves these.
If you only do one thing
Same number multiplied each step. The nth term is tn = t1 × r(n − 1). Find t₁ and r first.
The 10-minute version
Low energy? Do just this and you have still won the day. Zero is the only fail.
The one move: read the first formula box and the first worked example above, then do Practice Question 1. Screenshot the win for Nat and stop there.
📺 Watch it explained
Four short ways in. The 🎬 cards are waiting on Nat's videos, and each has a ready-to-read film script tucked underneath. 🖨️ Prefer printable notes? 4 styles here.
Section 1 · Spotting a Geometric Sequence
Divide any term by the one before it. If you get the same answer every time, it's geometric, and that answer is your common ratio r.
To find r: divide any term by the one before it. 6÷2 = 3, 18÷6 = 3, 54÷18 = 3. ✓
📈 Growth (r > 1)
The sequence gets bigger each step.
e.g. bacteria doubling: r = 2
e.g. investments: r = 1.05 (5% growth)
e.g. investments: r = 1.05 (5% growth)
📉 Decay (0 < r < 1)
The sequence gets smaller each step.
e.g. drug leaving body: r = 0.5
e.g. depreciation: r = 0.8 (20% loss)
e.g. depreciation: r = 0.8 (20% loss)
Section 2 · The nth Term Formula
tₙ = t₁ × r n−1
where t₁ = first term, r = common ratio, n = term position
tₙ
The nth term
The value at position n in the sequence. That's what you're usually solving for.
t₁
First term (t₁)
The starting value of the sequence. Sometimes called t₁ or just "the first term."
r
Common ratio
The multiplier. r > 1 → growth. 0 < r < 1 → decay. Always: r = t₂ ÷ t₁.
⚠️ The most common mistake: it's rⁿ⁻¹, not rⁿ. When n = 1 (first term), the power is zero, and r⁰ = 1, so tₙ = t₁×1 = t₁. That makes sense: the first term is just t₁. If you use rⁿ by mistake, every answer will be off by one factor of r.
Find tₙ
Given t₁, r, n
Substitute directly: tₙ = t₁ × rⁿ⁻¹
Find n
Given t₁, r, tₙ
Set up a × rⁿ⁻¹ = value. If r is a whole number, work out what power of r gives you the right answer.
Find t₁ or r
Given two terms
Divide tₙ by t₁ to get r raised to some power: r = ⁿ⁻¹√(tₙ ÷ t₁)
Section 3 · Worked Example
🌊 Coastal Café, Viral Post
Maria posts a photo of the new café fit-out. On day 1, 3 people share it. Each day, the number of new shares triples. She needs to know the numbers before pitching to a local sponsor.
Setup
Identify t₁, r, and write the general formula.
First term: t₁ = 3 (3 shares on day 1)
Common ratio: r = 3 (triples each day)
General formula: tₙ = 3 × 3ⁿ⁻¹
Common ratio: r = 3 (triples each day)
General formula: tₙ = 3 × 3ⁿ⁻¹
Part a · Find the number of shares on day 5
How many new shares does Maria's post get on day 5?
tₙ = t₁ × rⁿ⁻¹
t₅ = 3 × 3⁵⁻¹
t₅ = 3 × 3⁴
t₅ = 3 × 81
t₅ = 243 shares
t₅ = 3 × 3⁵⁻¹
t₅ = 3 × 3⁴
t₅ = 3 × 81
t₅ = 243 shares
Part b · Find which day first hits 729 shares
On which day does Maria first get 729 new shares?
Set tₙ = 729:
3 × 3ⁿ⁻¹ = 729
3ⁿ⁻¹ = 729 ÷ 3 = 243
3ⁿ⁻¹ = 3⁵ (since 3⁵ = 243)
n − 1 = 5
n = 6 → Day 6
3 × 3ⁿ⁻¹ = 729
3ⁿ⁻¹ = 729 ÷ 3 = 243
3ⁿ⁻¹ = 3⁵ (since 3⁵ = 243)
n − 1 = 5
n = 6 → Day 6
Section 4 · Practice Questions
Tap a question to reveal the full worked answer.
Question 1
A sequence begins: 2, 6, 18, 54, 162, ...
(a) What is the common ratio? (b) What is t₇?
(a) What is the common ratio? (b) What is t₇?
▼
(a) r = t₂ ÷ t₁ = 6 ÷ 2 = 3
(b) tₙ = t₁ × rⁿ⁻¹
t₇ = 2 × 3⁷⁻¹ = 2 × 3⁶ = 2 × 729 = 1458
(b) tₙ = t₁ × rⁿ⁻¹
t₇ = 2 × 3⁷⁻¹ = 2 × 3⁶ = 2 × 729 = 1458
✓ r = 3, t₇ = 1458
Question 2
A medication halves in the bloodstream every hour. The reading at t = 0 is 800 mg (call this t₁). How much remains after 4 hours? (That is t₅, since t₁ is the hour-0 reading. Use r = 0.5.)
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t₅ = 800 × (0.5)⁵⁻¹
t₅ = 800 × (0.5)⁴
t₅ = 800 × 0.0625
t₅ = 50 mg
t₅ = 800 × (0.5)⁴
t₅ = 800 × 0.0625
t₅ = 50 mg
✓ 50 mg remains 4 hours after the first reading. Because t₁ is the hour-0 reading, the value after 4 hours is the 5th term, t₅.
Question 3
A geometric sequence has first term t₁ = 4 and common ratio r = 3. Which term in the sequence equals 972?
▼
Set tₙ = 972:
4 × 3ⁿ⁻¹ = 972
3ⁿ⁻¹ = 972 ÷ 4 = 243
3ⁿ⁻¹ = 3⁵ (3⁵ = 243 ✓)
n − 1 = 5
n = 6
4 × 3ⁿ⁻¹ = 972
3ⁿ⁻¹ = 972 ÷ 4 = 243
3ⁿ⁻¹ = 3⁵ (3⁵ = 243 ✓)
n − 1 = 5
n = 6
✓ t₆ = 972 (it's the 6th term)
Question 4, Find r from two terms
A geometric sequence has t₁ = 7 and t₄ = 189. Find the common ratio r.
▼
t₄ = t₁ × r³ (since 4 − 1 = 3)
189 = 7 × r³
r³ = 189 ÷ 7 = 27
r = ∛27 = 3
Check: t₁=7, t₂=21, t₃=63, t₄=189 ✓
189 = 7 × r³
r³ = 189 ÷ 7 = 27
r = ∛27 = 3
Check: t₁=7, t₂=21, t₃=63, t₄=189 ✓
✓ r = 3
The Lab Report, Escape Room
A science competition at midnight. Your bacterial growth analysis needs to be verified before the submission portal closes. 6 challenges on geometric sequences.
Play →