IB Maths AA SL Topic 1 — Number & Algebra Paper 1 & 2 🎯 Skill ~4 min practice

AA SL Geometric Series Sums skills

A geometric series multiplies each term by the same ratio. There are two formulas — one for finite sums (S_n) and one for infinite sums (S_∞). The trick is choosing correctly and checking convergence before you reach for the infinite version.

The Method

GP: u₁, u₁r, u₁r², u₁r³, … u₁ = first term · r = common ratio · n = number of terms

Formula 1

Finite Sum S_n

S_n = u₁(rⁿ − 1)r − 1

use for the first n terms

Formula 2

Infinite Sum S_∞

S_∞ = u₁1 − r

use when sum is infinite · only if |r| < 1

⚠️

Infinite sum exists ONLY when |r| < 1

If |r| ≥ 1 the terms don’t shrink and the series doesn’t converge. Always state |r| < 1 before you apply S_∞ = u₁ / (1 − r).

Three steps every time

Identify what you have. Pick out u₁, r, and either n (finite) or “∞” (infinite). r = any term ÷ previous term.
Choose the formula. Finite “first n terms”? → Formula 1. “Sum to infinity”? → Check |r| < 1 first, then Formula 2.
Plug in carefully. Watch the order of (rⁿ − 1) and (r − 1) — both numerator and denominator have r minus 1, not 1 minus r.

Worked examples

WE 1 EASY

Find the sum of the first 8 terms of: 3, 6, 12, 24, …

step 1 — identify u₁ = 3, r = 6 ÷ 3 = 2, n = 8 step 2 — Formula 1 (finite) S₈ = 3(2⁸ − 1) / (2 − 1) = 3(256 − 1) / 1 = 3 × 255 S₈ = 765 r = (any term) ÷ (previous term) — pick any pair to find it!

WE 2 MEDIUM

Find the sum to infinity of: 12, 4, 4/3, 4/9, …

step 1 — identify u₁ = 12, r = 4 ÷ 12 = ⅓ step 2 — check convergence |⅓| < 1 ✓ → S∞ exists step 3 — Formula 2 S∞ = 12 / (1 − ⅓) = 12 / (⅔) = 12 × (3/2) = 18 S∞ = 18 always state |r| < 1 before applying S∞ — it’s a marking criterion!

WE 3 HARD

A geometric series has u₁ = 5 and r = 2. Find the smallest value of n for which S_n > 1000.

step 1 — set up the inequality 5(2ⁿ − 1) / (2 − 1) > 1000 5(2ⁿ − 1) > 1000 2ⁿ − 1 > 200 2ⁿ > 201 step 2 — solve using logs n log 2 > log 201 n > log 201 / log 2 ≈ 7.65 step 3 — round UP to next integer n must be a whole number ≥ 7.65 smallest n = 8 “smallest n such that S_n > k” — always round UP, never down!

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. Identify r first — that tells you which formula to use.

Q1 EASY Find the sum of the first 6 terms of: 2, 6, 18, 54, … Show answer ▼Hide answer ▲

u₁ = 2, r = 3, n = 6 → Formula 1 S₆ = 2(3⁶ − 1) / (3 − 1) = 2(729 − 1) / 2 = 728 S₆ = 728

Q2 EASY Find the sum to infinity of: 16, 8, 4, 2, … Show answer ▼Hide answer ▲

u₁ = 16, r = ½ — |r| < 1 ✓ S∞ = 16 / (1 − ½) = 16 / (½) S∞ = 32

Q3 MEDIUM A GP has u₁ = 80 and r = −½. Find S∞. Show answer ▼Hide answer ▲

|−½| = ½ < 1 ✓ → converges S∞ = 80 / (1 − (−½)) = 80 / (3/2) = 80 × (2/3) S∞ = 160/3 negative r is fine — it’s |r| that matters for convergence!

Q4 MEDIUM Find the sum of the first 10 terms of: 100, 50, 25, 12.5, … Show answer ▼Hide answer ▲

u₁ = 100, r = ½, n = 10 → Formula 1 S₁₀ = 100((½)¹⁰ − 1) / (½ − 1) = 100(1/1024 − 1) / (−½) = 100(−1023/1024) / (−½) = 200 × 1023/1024 = 199.8 (4 sf) S₁₀ ≈ 199.8 use the GDC for messy fractions — no shame in it on Paper 2!

Q5 HARD A GP has u₁ = 4 and S∞ = 10. Find r. Show answer ▼Hide answer ▲

step 1 — set up 10 = 4 / (1 − r) step 2 — solve for r 10(1 − r) = 4 10 − 10r = 4 10r = 6 r = 3/5 = 0.6 check: |0.6| < 1 ✓ → series does converge as required

⚠ Common mistakes

Using S_∞ when |r| ≥ 1. The infinite formula only works for convergent series. If |r| ≥ 1, the sum diverges — there’s no finite answer.
Picking the wrong formula. “Sum of first 10 terms” is finite (Formula 1). “Sum to infinity” needs Formula 2.
Confusing r with d. Geometric uses ratio (divide), arithmetic uses difference (subtract). Don’t mix the two.
Forgetting to check |r| < 1 before applying S_∞. State it explicitly — it’s part of the working.
Order in the formula. S_n = u₁(rⁿ − 1) / (r − 1). Both top and bottom have (r − 1), not (1 − r). The booklet has both forms — pick one and stay consistent.
Rounding down when “smallest n” is asked. If n > 7.65, the smallest integer is 8, not 7.

📖

Want the theory?

Read the full Geometric Sequences & Series notes for the link to u_n = u₁rⁿ⁻¹, why convergence requires |r| < 1, and applications to compound interest and depreciation.

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AA SL Geometric Series Sums skills

The Method

Finite Sum S_n

Infinite Sum S_∞

Infinite sum exists ONLY when |r| < 1

Three steps every time

Worked examples

Practice questions

⚠ Common mistakes

Need help with Geometric Series Sums?

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IB DEMYSTIFIED

AA SL Geometric Series Sums skills

The Method

Finite Sum Sn

Infinite Sum S∞

Infinite sum exists ONLY when |r| < 1

Three steps every time

Worked examples

Practice questions

⚠ Common mistakes

Need help with Geometric Series Sums?

Quick Links

Contact us

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Finite Sum S_n

Infinite Sum S_∞