The big idea: A geometric series goes on forever. But sometimes, all those infinite terms add up to a finite number.
The answer depends on one thing: is each term getting smaller and smaller toward zero?
The rule
| Common ratio r | What happens to the terms? | Does S∞ exist? |
|---|---|---|
| r = 2 (e.g. 2, 4, 8, 16, ...) | Terms grow bigger and bigger | ❌ No |
| r = 1 (e.g. 5, 5, 5, 5, ...) | Terms stay the same forever | ❌ No |
| r = −1 (e.g. 3, −3, 3, −3, ...) | Terms flip sign, never shrink | ❌ No |
| r = 0.5 (e.g. 8, 4, 2, 1, ...) | Terms shrink toward zero | ✅ Yes |
| r = −0.5 (e.g. 8, −4, 2, −1, ...) | Terms shrink in size (ignore sign) | ✅ Yes |
The S∞ condition: S∞ exists only when |r| < 1
|r| means the size of r, ignoring any negative sign. So r = −0.7 → |r| = 0.7 → |r| < 1 ✓
If |r| ≥ 1 → write "S∞ does not exist". Never calculate it. You will lose marks if you do.
Practice: does S∞ exist?
Series: 3 + 6 + 12 + 24 + ...
r = 6 ÷ 3 = 2 → |r| = 2 ≥ 1
S∞ does not exist. The terms keep growing.
Series: 10 + 5 + 2.5 + 1.25 + ...
r = 5 ÷ 10 = 0.5 → |r| = 0.5 < 1 ✓
S∞ exists.
Series: 6 − 4 + 8/3 − 16/9 + ...
r = −4 ÷ 6 = −2/3 → |r| = 2/3 < 1 ✓
S∞ exists. Negative r is fine as long as |r| < 1.
Series: 5 + 5 + 5 + 5 + ...
r = 5 ÷ 5 = 1 → |r| = 1 → NOT less than 1
S∞ does not exist. The terms never shrink.
The formula: Once you confirm |r| < 1, use:
S∞ = u₁ ÷ (1 − r)
You always need two things: the first term u₁, and the common ratio r.
How to remember this (not on the formula sheet): S∞ is not given in the IB formula booklet — only Sₙ is.
But you can build it from the formula you do have on your sheet:
Sₙ = u₁(1 − rⁿ) ÷ (1 − r)
When |r| < 1, rⁿ shrinks to zero as n → ∞. So just replace rⁿ with 0:
S∞ = u₁(1 − 0) ÷ (1 − r) = u₁ ÷ (1 − r)
You are not memorising a new formula — you are crossing out rⁿ from the one already on your sheet.
Type 1 — Find S∞ from a sequence
The sequence 8, 4, 2, 1, ... goes on forever.
Does S∞ exist?
If so, find it.
Step by step
- Find r: r = 4 ÷ 8 = 0.5
- Check: |0.5| < 1 ✓ — S∞ exists
- Write the formula: S∞ = u₁ ÷ (1 − r)
- Substitute: S∞ = 8 ÷ (1 − 0.5) = 8 ÷ 0.5
- S∞ = 16
Final answer
S∞ = 16
Type 2 — Find u₁ given S∞ and r
A geometric series has r = 0.4 and S∞ = 25.
Find the first term u₁.
Step by step
- Check: |0.4| < 1 ✓ (we're told S∞ = 25, so it already exists)
- Write the formula: S∞ = u₁ ÷ (1 − r)
- Substitute: 25 = u₁ ÷ (1 − 0.4)
- Simplify the bracket: 25 = u₁ ÷ 0.6
- Multiply both sides by 0.6: u₁ = 25 × 0.6
- u₁ = 15
Final answer
u₁ = 15
Type 3 — Find r given S∞ and u₁
A geometric series has u₁ = 12 and S∞ = 20.
Find r.
Step by step
- Write the formula: S∞ = u₁ ÷ (1 − r)
- Substitute: 20 = 12 ÷ (1 − r)
- Rearrange: 1 − r = 12 ÷ 20 = 0.6
- r = 1 − 0.6
- r = 0.4
Final answer
r = 0.4
Rearranging the formula: The formula S∞ = u₁ ÷ (1 − r) has three variables.
If you know any two, you can always find the third:
- Find S∞: substitute u₁ and r directly - Find u₁: multiply both sides by (1 − r) - Find r: substitute S∞ and u₁, then solve 1 − r = u₁ ÷ S∞
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Two ideas, one question: IB questions on this topic often come in two parts.
- 'Will it ever reach the target?' → use S∞. Calculate S∞ and compare it to the target. If S∞ < target, it can never get there. - 'Reach the target after n steps?' → use Sₙ. Set Sₙ = target and solve for the first deposit.
The signal: - 'never reach' / 'cannot reach' → S∞ - 'after n years / weeks / steps' → Sₙ
Practice Question — 6 marks: Leila is saving money for a new laptop. At the start of each week, she puts money into a savings tin. In the first week she deposits $6 000. Each week after that, she adds exactly half the amount she deposited the week before.
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(a) Show that Leila will never reach her savings target of $15 000, no matter how many weeks pass. [3 marks]
(b) Leila wants to reach exactly $15 000 after 4 weeks of saving. Find the minimum amount she needs to deposit in the first week. Give your answer to the nearest dollar. [3 marks]
Part (a) — Show it never reaches the target
If the first deposit is 6 000 and r = 0.5, show that the total saved can never reach $15 000.
Step by step
- Identify the structure: each week adds half of the previous week — this is a geometric series
- Common ratio: r = 0.5 (each term is half the previous one)
- Check: |0.5| < 1 ✓ — S∞ exists
- Maximum possible total (saving forever): S∞ = first deposit ÷ (1 − r)
- Substitute: S∞ = 6000 ÷ (1 − 0.5) = 6000 ÷ 0.5
- S∞ = 12 000
- Since 12 000 < 15 000, even saving forever will not reach the target
- Therefore, Leila will never reach $15 000. ✓
Final answer
S∞ = 12 000 < 15 000 — target is never reached.
What 'Show that...' means: The answer is printed in the question — your job is to prove it.
You need three things: 1. Calculate S∞ and write the number: S∞ = 12 000 2. Compare it to the target: 12 000 < 15 000 3. Write a conclusion in words: 'Even saving forever, Leila cannot reach $15 000.'
Most students stop at step 1. That loses a mark.
Part (b) — Find the minimum initial deposit
Find the first deposit so that the total saved after exactly 4 weeks equals $15 000.
Step by step
- This is a finite sum — we are saving for exactly 4 weeks, not forever
- Use the Sₙ formula: Sₙ = first deposit × (1 − rⁿ) ÷ (1 − r)
- Set S₄ = 15 000 with r = 0.5 and n = 4
- 15 000 = first deposit × (1 − 0.5⁴) ÷ (1 − 0.5)
- 15 000 = first deposit × (1 − 0.0625) ÷ 0.5
- 15 000 = first deposit × 0.9375 ÷ 0.5
- 15 000 = first deposit × 1.875
- first deposit = 15 000 ÷ 1.875
- first deposit = 8 000
Final answer
Minimum initial deposit = $8 000
Two questions, two different formulas: IB questions on this topic often come in two parts — back to back.
'Will it ever reach the target?' → use S∞ Calculate S∞. If S∞ < target, it can never get there.
'Reach the target after n steps?' → use Sₙ Set Sₙ = target and solve for the first deposit.
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The signal words: - never reach / cannot reach → S∞ - after n years / weeks / steps → Sₙ
What to write to score full marks: Part (a) — Show it never reaches the target: - Write r = 0.5 in your working — this one step earns a mark on its own - Use the S∞ formula and write the result (S∞ = 12 000) - Write the comparison: 12 000 < 15 000 - Write a conclusion sentence: 'Even saving forever, Leila cannot reach $15 000.'
Leaving out the comparison or conclusion sentence loses a mark — the number alone is not enough.
Part (b) — Find the first deposit: - Write the equation with the total set equal to 15 000 — this earns a mark even before you solve it - Show the unrounded answer - Give the final answer rounded to the nearest dollar
On both parts: always write the equation or formula before your calculator answer. A correct number with no working shown does not score full marks.