IB Maths
Paper 1 & 2
22 min read
Geometric Sequences & Series
A geometric sequence multiplies (or divides) by the same number each time. Three formulas cover almost every IB question — including the famous sum to infinity. Let’s go.
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What you need to know
- How to spot a geometric sequence (look for a constant ratio)
- The nth term formula using u1 and r
- The sum formula for the first n terms
- The sum to infinity (only when |r| < 1)
- What “converge” and “diverge” mean
What is a geometric sequence?
A sequence where each term is multiplied by the same number — called the common ratio, r.
Quick examples:
- 2, 6, 18, 54, 162, … r = 3 (increasing)
- 100, 50, 25, 12.5, … r = 12 (decreasing)
- 3, −6, 12, −24, … r = −2 (alternating signs)
- 1000, 100, 10, 1, … r = 110
How to find r: divide any term by the one before it.
r = u2u1 = u3u2 = un+1un
Three types of behaviour
- Increasing when r > 1 (e.g. 2, 6, 18, 54, …)
- Decreasing when 0 < r < 1 (e.g. 100, 50, 25, …)
- Alternating when r < 0 (signs flip each time)
Formula 1: The nth term
un = u1 rn − 1
Where:
- u1 = first term
- r = common ratio
- n = position of the term
This formula is in the IB formula booklet.
Quick example: find u8 for the sequence 2, 6, 18, 54, …
- u1 = 2, r = 3
- u8 = 2 × 38 − 1 = 2 × 37
- u8 = 2 × 2187 = 4374
Formula 2: The sum of the first n terms
You get two versions in the formula booklet:
Sn = u1(rn − 1)r − 1
— easier when r > 1
Sn = u1(1 − rn)1 − r
— easier when r < 1
Quick example: sum of 2 + 6 + 18 + 54 + … up to 8 terms
- u1 = 2, r = 3, n = 8
- Use the first version (r > 1):
- S8 = 2(38 − 1)3 − 1 = 2(6561 − 1)2 = 6560
Formula 3: The sum to infinity
If you keep adding terms forever, what do you get? Two cases:
- If |r| ≥ 1: the sum grows forever → diverges (no finite answer)
- If |r| < 1: the sum gets closer and closer to a finite value → converges
For example:
- 1 + 2 + 4 + 8 + 16 + … r = 2 → diverges to ∞
- 1 + 12 + 14 + 18 + … r = 12 → converges to 2
When |r| < 1, you can use this formula:
S∞ = u11 − r (only when |r| < 1)
Both the formula and the condition |r| < 1 are in the formula booklet. Always state the condition in your answer.
Quick example: u1 = 1, r = 12
- Check |r| < 1: |12| = 12 < 1 ✓ (converges)
- S∞ = 11 − 12 = 112 = 2
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Tips
- To find r from two terms, divide them: r = u7u6
- To find n when given a term, you’ll often need logarithms
- Always check |r| < 1 before using S∞ — otherwise the formula doesn’t apply
Worked Examples
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Example 1 — Find a term using u1 and r
A geometric sequence has first term 25 and common ratio 0.8. Find u5 and S5.
Answer:
Find u5:
u5 = 25 × (0.8)4
= 25 × 0.4096
u5 = 10.24
Find S5 (use second formula since r < 1):
S5 = 25(1 − 0.85) / (1 − 0.8)
= 25(1 − 0.32768) / 0.2
= 25(0.67232) / 0.2
u5 = 10.24, S5 = 84.04
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Example 2 — Find r and u1 from two terms
The 6th term of a geometric sequence is 486 and the 7th term is 1458. Find r and u1.
Answer:
Find r by dividing consecutive terms:
r = u7 / u6 = 1458 / 486 = 3
Find u1 using un = u1rn − 1:
u6 = u1(3)5 = 486
243 u1 = 486
r = 3, u1 = 2
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Example 3 — Find n using logarithms
A geometric sequence has u1 = 5 and r = 2. Find the smallest value of n for which un > 10000.
Answer:
Set up the inequality:
5 × 2n − 1 > 10000
2n − 1 > 2000
Take logs of both sides:
(n − 1) log 2 > log 2000
n − 1 > log(2000) / log(2)
n − 1 > 10.965…
n > 11.965…
Smallest whole n:
n = 12
Check: u12 = 5 × 211 = 10240 ✓
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Example 4 — Sum to infinity
The first three terms of a geometric sequence are 6, 2, 23. Show that the series converges and find S∞.
Answer:
Find r:
r = u2 / u1 = 2/6 = 1/3
Check convergence:
|1/3| < 1 ✓ series converges
Apply S∞ formula:
S∞ = 6 / (1 − 1/3) = 6 / (2/3) = 9
S∞ = 9
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Example 5 — Find u1 from S∞
A geometric series has r = 0.4 and S∞ = 25. Find the first term.
Answer:
Use S∞ = u1 / (1 − r):
25 = u1 / (1 − 0.4)
25 = u1 / 0.6
u1 = 25 × 0.6
u1 = 15
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Common mistakes
- Confusing d (arithmetic) with r (geometric). Arithmetic adds, geometric multiplies.
- Forgetting the (n − 1) exponent. The formula is u1 rn − 1, NOT u1 rn.
- Using S∞ when |r| ≥ 1. The formula only works when |r| < 1. Always check first.
- Wrong sign for r. If terms alternate signs (e.g. 3, −6, 12, −24), r is negative.
- Mixing up the two sum formulas. Use the one that keeps the brackets positive: r > 1 → first version, r < 1 → second version.
- Dividing the wrong way. r = (next term) ÷ (previous term), NOT the other way around.
Final word: 3 formulas, all in the booklet. The trickiest part is knowing which one to use — practise spotting whether you need un, Sn, or S∞. For “find n” questions, get comfortable with logarithms.
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