A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number called the common ratio. From compound interest to bouncing balls, geometric sequences model real-world exponential growth and decay.
If you know uₘ and uₙ (not necessarily consecutive), use:
uₙ / uₘ = r^(n−m) → solve for r
Example: u₂ = 6 and u₅ = 48
u₅ / u₂ = r^(5−2) = r³
48/6 = 8 = r³ → r = 2
Then a = u₁ = u₂/r = 6/2 = 3
Recognising Geometric vs Arithmetic
Feature
Arithmetic
Geometric
Pattern
Add constant d
Multiply by constant r
Test
u₂ − u₁ = u₃ − u₂
u₂ ÷ u₁ = u₃ ÷ u₂
nth term
a + (n−1)d
a · r^(n−1)
Graph shape
Linear (straight line)
Exponential (curve)
Example
2, 5, 8, 11, ...
2, 6, 18, 54, ...
Tip: Check if ratios are constant (geometric) OR differences are constant (arithmetic). A sequence that multiplies by the same number each time is geometric — even if r is a fraction or negative.
Learn 2 — Sum of a Geometric Series
Sum of First n Terms
The sum of the first n terms of a geometric series is:
Sₙ = a(rⁿ − 1) / (r − 1) when r ≠ 1
An equivalent form (useful when r < 1 to avoid negative numerator/denominator):
Sₙ = a(1 − rⁿ) / (1 − r) when r ≠ 1
Example: Find S₆ for 2, 6, 18, 54, ...
a = 2, r = 3, n = 6
S₆ = 2(3⁶ − 1)/(3 − 1) = 2(729 − 1)/2 = 728
Example (r < 1): Find S₄ for 80, 40, 20, 10, ...
a = 80, r = 0.5, n = 4
S₄ = 80(1 − 0.5⁴)/(1 − 0.5) = 80(1 − 0.0625)/0.5 = 80 × 0.9375/0.5 = 150
Sum to Infinity (|r| < 1)
When the common ratio satisfies |r| < 1, each term gets smaller and the series converges to a finite sum:
S∞ = a / (1 − r) [only valid when |r| < 1]
Why does this work? When |r| < 1, as n → ∞, rⁿ → 0.
Sₙ = a(1 − rⁿ)/(1 − r) → a(1 − 0)/(1 − r) = a/(1 − r)
Example: 1 + ½ + ¼ + ⅛ + ...
a = 1, r = 0.5 (|r| < 1 ✓)
S∞ = 1/(1 − 0.5) = 1/0.5 = 2
Warning: If |r| ≥ 1, the series DIVERGES — the sum to infinity does NOT exist. E.g. r = 2: terms keep growing → no finite sum.
Tip: Always check |r| < 1 before using S∞ = a/(1−r). If r = −0.5, |r| = 0.5 < 1, so it works (alternating converging series).
Learn 3 — Applications & Comparison with Arithmetic
Compound Interest
Money invested at compound interest is a geometric sequence:
Amount after n years = P × rⁿ where r = 1 + (rate/100)
Example: £1000 invested at 5% p.a. compound interest
P = 1000, r = 1.05
Year 1: 1050, Year 2: 1102.50, Year 3: 1157.63, ...
After 10 years: 1000 × 1.05¹⁰ = £1628.89 (use uₙ = ar^(n−1) with a = 1000r, or simply P×rⁿ)
Key distinction: Use uₙ = ar^(n−1) for the VALUE at a specific time (single term). Do NOT use Sₙ for compound interest — Sₙ is the TOTAL of all terms added together.
Population Growth and Decay
Growth: Population of 5000 grows at 3% per year.
After n years: 5000 × 1.03ⁿ
After 20 years: 5000 × 1.03²⁰ ≈ 9031
Decay: Radioactive substance with half-life = each year r = 0.5
After n years: A × (0.5)ⁿ
Bouncing Ball
Ball dropped from height 10 m, each bounce reaches 60% of previous height.
Heights: 10, 6, 3.6, 2.16, ... → geometric with a=10, r=0.6
Height after nth bounce: uₙ = 10 × (0.6)^(n−1)
Total distance travelled ≈ 2 × S∞ − a = 2 × 10/(1−0.6) − 10 = 50 − 10 = 40 m (accounts for up and down)
Arithmetic vs Geometric — Full Comparison
Property
Arithmetic
Geometric
Pattern
Add d each time
Multiply by r each time
nth term
a + (n−1)d
ar^(n−1)
Sum of n terms
n/2 × (2a + (n−1)d)
a(rⁿ−1)/(r−1)
Graph
Linear (straight line)
Exponential (curve)
Sum to infinity
Always diverges
Converges if |r|<1
Real-world model
Simple interest, steady salary
Compound interest, population
When to Use Which Formula
Finding a specific term → use uₙ = ar^(n−1)
Finding total accumulated over n periods → use Sₙ
Finding long-run total (infinitely many terms) → use S∞ (check |r|<1 first!)
Finding which term equals a value → set ar^(n−1) = value, solve using logarithms or trial
Worked Examples
Example 1 — Find the nth Term
Find the nth term of the geometric sequence: 5, 15, 45, 135, ...
r = 15 ÷ 5 = 3 M1
a = 5 (first term)
uₙ = 5 × 3^(n−1) A1
Check: u₄ = 5 × 3³ = 5 × 27 = 135 ✓
Example 2 — Sum of First n Terms
Find the sum of the first 8 terms of: 2, 6, 18, 54, ...
a = 2, r = 3, n = 8 B1
S₈ = 2(3⁸ − 1)/(3 − 1) M1
3⁸ = 6561
S₈ = 2(6561 − 1)/2 = 2 × 6560/2 = 6560 A1
Example 3 — Sum to Infinity
Find the sum to infinity of: 8 + 4 + 2 + 1 + ...
a = 8, r = 4/8 = 0.5, |r| = 0.5 < 1 ✓ (series converges) B1
S∞ = a/(1−r) = 8/(1−0.5) M1
S∞ = 8/0.5 = 16A1
Example 4 — Compound Interest
£500 is invested at 4% compound interest per year. Find the value after 6 years.
This is geometric with a = 500, r = 1.04 M1
Value after 6 years = 500 × 1.04⁶ M1
= 500 × 1.26532... = £632.66 (2 d.p.) A1
Note: We use uₙ (the nth term), NOT Sₙ. We want the amount, not the total of all years added together.
Example 5 — Find a and r from Two Terms
A geometric sequence has u₃ = 12 and u₆ = 96. Find a and r.
u₆/u₃ = r^(6−3) = r³ M1
96/12 = 8 → r³ = 8 → r = 2 A1
u₃ = ar² = 12 → a × 4 = 12 → a = 3 A1
Check: u₁=3, u₂=6, u₃=12 ✓, u₆=3×2⁵=96 ✓
Example 6 — Prove Convergence & Find S∞
The first term of a geometric sequence is 20 and the fourth term is 2.5. Show the series converges and find S∞.
u₄ = ar³ = 2.5, a = 20 M1
r³ = 2.5/20 = 0.125 = (0.5)³ → r = 0.5 A1
|r| = 0.5 < 1, therefore the series converges. B1
S∞ = 20/(1−0.5) = 20/0.5 = 40A1
Common Mistakes
Here are the most frequent errors students make — learn these so you don't make them!
Mistake 1 — Using the Arithmetic nth Term for a Geometric Sequence
✓ Check whether to add or multiply. Ratios are constant (×2 each time), so it's geometric: uₙ = 3 × 2^(n−1). The arithmetic formula a+(n−1)d only works when you're ADDING the same number each time.
✓ Always check |r| < 1 first. Here r = 3, |r| = 3 ≥ 1, so the series DIVERGES. The sum to infinity does not exist. The formula S∞ = a/(1−r) only applies when |r| < 1.
Mistake 3 — Confusing uₙ (nth Term) with Sₙ (Sum)
Question asks: "Find the 6th term." Student calculates S₆ instead of u₆ ✗
✓ uₙ = ar^(n−1) gives the VALUE of the nth term. Sₙ = a(rⁿ−1)/(r−1) gives the TOTAL of the first n terms added together. These are completely different values!
Mistake 4 — Off-By-One Error in the Exponent
"Find the 5th term": student writes u₅ = ar⁵ instead of ar⁴ ✗
✓ The formula is uₙ = ar^(n−1). For the 5th term: n=5, so exponent = 5−1 = 4. Reason: the first term has r⁰ = 1 (no multiplication yet), second has r¹, third r², etc.
Mistake 5 — Using Sₙ for Compound Interest
£1000 at 5% for 6 years. Student calculates S₆ = sum of all yearly amounts ✗
✓ Compound interest asks for the AMOUNT at a specific time — this is a single term, not a sum. Use: Amount = P × rⁿ = 1000 × 1.05⁶. Sₙ would give the total of all years' balances added together — not what the question asks.
Key Formulas
Know which formulas are given in the exam and which you must memorise.
Formula
What it Does
Status
uₙ = ar^(n−1)
nth term of geometric sequence
Must Memorise
Sₙ = a(rⁿ−1)/(r−1)
Sum of first n terms (r≠1)
Must Memorise
Sₙ = a(1−rⁿ)/(1−r)
Equivalent form (useful r<1)
Must Memorise
S∞ = a/(1−r)
Sum to infinity (|r|<1 only)
Must Memorise
r = u₂/u₁
Finding the common ratio
Derive Easily
Amount = P×rⁿ
Compound interest after n years
Must Memorise
uₙ = a+(n−1)d (arithmetic)
nth term — arithmetic only
Must Memorise
Sₙ = n/2(2a+(n−1)d) (arithmetic)
Sum — arithmetic only
Given in Exam
Key conditions to state: For S∞, always write "|r| < 1" as your justification that the series converges. Examiners award a mark for this!
Quick Reference — Geometric Identities:
• r > 1: sequence grows → diverges
• 0 < r < 1: sequence decreases toward 0 → converges
• r < 0: sequence alternates sign → converges if |r| < 1
• r = 1: all terms equal a → Sₙ = na
• r = −1: alternates a, −a, a, ... → sum oscillates, no limit
Geometric Sequence Visualiser
Enter a and r to see the first 10 terms plotted as a bar chart. See partial sums and convergence if |r|<1.
Exercise 1 — Finding the nth Term (8 Questions)
Exercise 2 — Find the Term Number (8 Questions)
Exercise 3 — Sum of n Terms (8 Questions)
Exercise 4 — Sum to Infinity (8 Questions)
Exercise 5 — Applications (8 Questions)
Practice — 25 Questions
🔵 = Non-calculator 🟢 = Calculator allowed
Challenge — 12 Multi-Step Questions
Exam Style Questions
Structured questions in Cambridge IGCSE format. Show all working to gain method marks.
Question 1 (5 marks)
A geometric sequence has first term a = 6 and common ratio r = 2.
(a) Write down the first four terms. [1 mark]
(b) Find the 10th term. [2 marks]
(c) Find the sum of the first 10 terms. [2 marks]
Question 2 (5 marks)
A geometric series has first term 24 and sum to infinity 96.
(a) Find the common ratio r. [2 marks]
(b) Find the 5th term. [2 marks]
(c) Find the sum of the first 6 terms. [1 mark]
Question 3 (4 marks)
The 2nd term of a geometric sequence is 10 and the 5th term is 80.
(a) Find the common ratio. [2 marks]
(b) Find the first term. [1 mark]
(c) Find the 8th term. [1 mark]
Question 4 (3 marks)
£2000 is invested at 3.5% compound interest per year. Find the value after 5 years, correct to the nearest pound.
[3 marks]
Question 5 (4 marks)
A ball is dropped from a height of 16 m. Each bounce reaches 75% of the previous height.
(a) Find the height reached after the 4th bounce. [2 marks]
(b) Find the total distance the ball travels (up and down, as it eventually stops). Use S∞. [2 marks]