The big idea: A geometric sequence is a list of numbers where you multiply to get the next one.
You multiply by the same amount every time.
That amount has a name — we call it r.
Let's see it
Look at this sequence: 2, 6, 18, 54, …
- 2 × 3 = 6
- 6 × 3 = 18
- 18 × 3 = 54
Every step you multiply by 3.
So r = 3.
That's it — that's a geometric sequence.
Geometric vs arithmetic
Arithmetic sequences add the same number each time.
Geometric sequences multiply by the same number each time.
| Type | What you do each step | Example |
|---|---|---|
| Arithmetic | Add the same number | 2, 5, 8, 11, … (add 3) |
| Geometric | Multiply by the same number | 2, 6, 18, 54, … (multiply by 3) |
Quick way to tell them apart: Look at how you go from one term to the next.
If you add the same number → arithmetic.
If you multiply by the same number → geometric.
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The big idea: To find r, divide any term by the term before it.
The ratio stays the same throughout a geometric sequence.
| Part | Meaning |
|---|---|
| r | the common ratio |
| any term in the sequence | |
| the term right after it |
Worked example — find r
Find r for the sequence 5, 15, 45, 135, …
Step by step
- Take the second term and divide by the first.
- Check with another pair to make sure the ratio stays the same.
Final answer
So the common ratio is 3.
What r tells you
| Value of r | What happens | Example |
|---|---|---|
| r > 1 | Terms grow | 2, 6, 18, 54, … (r = 3) |
| 0 < r < 1 | Terms shrink | 80, 40, 20, 10, … (r = 0.5) |
| r < 0 | Signs alternate | 4, −8, 16, −32, … (r = −2) |
| r = 1 | Terms stay the same | 5, 5, 5, 5, … |
Common mistake: Do not subtract to find r.
In a geometric sequence, you divide consecutive terms.
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The big idea: To find any term in a geometric sequence without listing them all, use the nth-term rule.
What each part means
| Part | Meaning |
|---|---|
| uₙ | The term you want (the nth term) |
| u₁ | The first term |
| r | The common ratio |
| n | The position (1st, 2nd, 3rd, …) |
| rⁿ⁻¹ | r multiplied by itself (n − 1) times |
The exponent is n − 1, not n.
The first term needs zero multiplications by r — it is already u₁.
Worked example 1 — find u₈
Sequence: 3, 6, 12, 24, … Find u₈.
Step by step
- Find u₁ and r.
- Substitute n = 8 into the formula.
- Evaluate.
Final answer
So the 8th term is 384.
Worked example 2 — find u₅
Sequence: 80, 40, 20, … Find u₅.
Step by step
- Find u₁ and r.
- Substitute n = 5 into the formula.
- Evaluate.
- Check by listing the terms.
Final answer
So the 5th term is 5.
Exam tip: This is a very high-frequency exam style.
Always write the formula step before plugging in numbers — it earns method marks even if your final number is wrong.
The big idea: Sometimes the IB gives you a term value and asks which position it is.
In that case, you are finding n.
Type 1 — find n when uₙ is known
Sequence 3, 6, 12, 24, … Which term equals 384?
Step by step
- First identify the first term and the common ratio.
- Write what else you know.
- Substitute into uₙ = u₁·rⁿ⁻¹.
- Divide both sides by 3 to isolate the power.
- Recognise that 128 = 2⁷.
- Rewrite the equation using the same base on both sides.
- When the bases are the same, the exponents must be equal.
- Add 1 to both sides to solve for n.
Final answer
384 is the 8th term, so n = 8.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
GDC tip — when the power is difficult: Whenever you cannot easily tell what power gives the answer, use the GDC equation solver instead of guessing.
Casio fx-CG50: MENU → EQUA → SOLVE works the same way.
Examiner trap — n vs uₙ: In exams: n is the position of the term, uₙ is the value.
Do not mix them up.
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The big idea: Other times the IB gives you a position and a later term, then asks for the first term u₁.
Use the same formula uₙ = u₁ · rⁿ⁻¹, substitute what you know, and solve for u₁.
Type 2 — find u₁ when r and uₙ are known
The 4th term of a geometric sequence is 54 and r = 3.
Find u₁.
Step by step
- Write what you know.
- Substitute into uₙ = u₁·rⁿ⁻¹.
- Simplify the power.
- Divide both sides by 27 to solve for u₁.
- Check by listing the sequence.
Final answer
u₁ = 2
Quick check: If your u₁ looks strange, list the first few terms to check that they really build to the term the question gave you.