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.
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 = a × r n−1
where a = 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.
a
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 = a×1 = a. That makes sense: the first term is just a. If you use rⁿ by mistake, every answer will be off by one factor of r.
Find tₙ
Given a, r, n
Substitute directly: tₙ = a × rⁿ⁻¹
Find n
Given a, 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 a or r
Given two terms
Divide tₙ by t₁ to get r raised to some power: r = ⁿ⁻¹√(tₙ ÷ a)
Section 3 · The Sum Formula
Total of the first n terms
Sₙ = a(rⁿ − 1) ÷ (r − 1)
Use this when r > 1 (growing sequence). The formula gives the total of all terms from t₁ to tₙ.
Sₙ = a(1 − rⁿ) ÷ (1 − r)
Exactly the same formula, just rearranged for when 0 < r < 1 (decaying sequence) so you don't get a negative denominator.
💡 Tip: Both forms give the same answer, use whichever avoids negatives. Most exam questions use r > 1, so the first form is more common.
Section 4 · 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 a, r, and write the general formula.
First term: a = 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 = a × 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
Part c · Total shares over first 4 days
What is the total number of shares across days 1 to 4?
S₄ = a(r⁴ − 1) ÷ (r − 1)
S₄ = 3(3⁴ − 1) ÷ (3 − 1)
S₄ = 3(81 − 1) ÷ 2
S₄ = 3 × 80 ÷ 2
S₄ = 240 ÷ 2
S₄ = 120 shares total
S₄ = 3(3⁴ − 1) ÷ (3 − 1)
S₄ = 3(81 − 1) ÷ 2
S₄ = 3 × 80 ÷ 2
S₄ = 240 ÷ 2
S₄ = 120 shares total
Section 5 · 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 = a × rⁿ⁻¹
t₇ = 2 × 3⁷⁻¹ = 2 × 3⁶ = 2 × 729 = 1458
(b) tₙ = a × 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. At t = 0 there is 800 mg. How much remains after 5 hours? (This is t₅ with a = 800 and r = 0.5.)
▼
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 after 4 hours of decay (from the initial reading)
Question 3
A geometric sequence has first term a = 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, Sum formula
A population of fish starts at 10 and triples every year. What is the total population count across the first 5 years? (Find S₅.)
▼
a = 10, r = 3, n = 5
S₅ = a(r⁵ − 1) ÷ (r − 1)
S₅ = 10(3⁵ − 1) ÷ (3 − 1)
S₅ = 10(243 − 1) ÷ 2
S₅ = 10 × 242 ÷ 2
S₅ = 2420 ÷ 2 = 1210
S₅ = a(r⁵ − 1) ÷ (r − 1)
S₅ = 10(3⁵ − 1) ÷ (3 − 1)
S₅ = 10(243 − 1) ÷ 2
S₅ = 10 × 242 ÷ 2
S₅ = 2420 ÷ 2 = 1210
✓ S₅ = 1210 total fish counted across 5 years
Question 5, Find r from two terms
A geometric sequence has t₁ = 7 and t₄ = 189. Find the common ratio r.
▼
t₄ = a × 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.
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