IB Math AA SL Topic 1.8: Infinite geometric series | Notes & Flashcards | Aimnova

IB Math AA SL — Infinite geometric series

Topic 1.8 of IB Mathematics: Analysis and Approaches covers Infinite geometric series, which is part of Unit 1: Number & Algebra. Students explore key concepts including Sum to infinity. A strong understanding of infinite geometric series is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Infinite geometric series

Key Idea: Add a geometric sequence forever and the total can settle to a finite number. It shows up on both papers, and the whole topic hinges on one convergence check.

♾️ The sum to infinity

S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1

$u_1$: the first term
$r$: the common ratio (next ÷ current)

S∞ exists only when |r| < 1, because the terms shrink toward 0. If |r| ≥ 1 the terms don't shrink, the total grows without limit, and there is no sum to infinity — give a finite sum instead. State |r| < 1 before you compute.

🔁 The exam variations

✏️ IB-style worked examples

IB-style question — find the sum to infinity

Find the sum to infinity of 18 + 6 + 2 + … .

Step by step:

Find r by dividing consecutive terms, and check it converges.
$r = \frac{6}{18} = \tfrac{1}{3} \quad (|r| < 1\ \checkmark)$
Substitute u₁ = 18 and r = ⅓ into the formula.
$S_\infty = \frac{18}{1 - \tfrac{1}{3}} = \frac{18}{\tfrac{2}{3}}$
Finish.
$= 18 \times \tfrac{3}{2} = 27$

Final answer:

S∞ = 27.

IB-style question — given S∞, find the first term

A geometric series has common ratio r = 0.4 and a sum to infinity of 45. Find the first term.

Step by step:

Write the formula and substitute what you know.
$45 = \frac{u_1}{1 - 0.4}$
Work out the denominator (1 − r).
$45 = \frac{u_1}{0.6}$
Multiply both sides by 0.6 to isolate u₁.
$u_1 = 45 \times 0.6 = 27$

Final answer:

u₁ = 27.

IB-style question — total distance of a bouncing ball

A ball is dropped from 12 m and rebounds to ½ of its height each bounce, forever. Find the total distance it travels.

Step by step:

It drops 12 m once; then each rebound is travelled up and back down. The rebound heights 6, 3, 1.5, … are geometric (u₁ = 6, r = ½) — sum them to infinity.
$S_\infty = \frac{6}{1 - \tfrac{1}{2}} = 12$
Total = the drop, plus twice the rebound sum.
$12 + 2(12) = 36$

Final answer:

36 m. (Shortcut: total = x(1 + r)/(1 − r); with r = ⅔ that's the classic δ = 5x.)

Important: S∞ = u₁/(1 − r) only works when |r| < 1. If |r| ≥ 1 there is no sum to infinity — the question wants a finite sum Sₙ (often the first 2m terms). Always check |r| before reaching for S∞.

Tap each card to reveal the answer.

Exam Tips

S∞ exists ONLY when |r| < 1; then S∞ = u₁/(1 − r). State the check first.
Find r as next ÷ current before doing anything else.
Given S∞, rearrange for u₁ or r — the denominator is 1 − r, never r.
Partial sums approach S∞: the gap is u₁rⁿ/(1 − r); set it below the tolerance and round up.
If |r| ≥ 1 there is no S∞ — give a finite sum Sₙ, simplifying with r²ᵐ = (r²)ᵐ.

IB Math AA SL — Infinite geometric series

Key concepts in Infinite geometric series

Key Idea: Add a geometric sequence forever and the total can settle to a finite number. It shows up on both papers, and the whole topic hinges on one convergence check.

♾️ The sum to infinity

S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1

$u_1$: the first term
$r$: the common ratio (next ÷ current)

S∞ exists only when |r| < 1, because the terms shrink toward 0. If |r| ≥ 1 the terms don't shrink, the total grows without limit, and there is no sum to infinity — give a finite sum instead. State |r| < 1 before you compute.

🔁 The exam variations

✏️ IB-style worked examples

IB-style question — find the sum to infinity

Find the sum to infinity of 18 + 6 + 2 + … .

Step by step:

Find r by dividing consecutive terms, and check it converges.
$r = \frac{6}{18} = \tfrac{1}{3} \quad (|r| < 1\ \checkmark)$
Substitute u₁ = 18 and r = ⅓ into the formula.
$S_\infty = \frac{18}{1 - \tfrac{1}{3}} = \frac{18}{\tfrac{2}{3}}$
Finish.
$= 18 \times \tfrac{3}{2} = 27$

Final answer:

S∞ = 27.

IB-style question — given S∞, find the first term

A geometric series has common ratio r = 0.4 and a sum to infinity of 45. Find the first term.

Step by step:

Write the formula and substitute what you know.
$45 = \frac{u_1}{1 - 0.4}$
Work out the denominator (1 − r).
$45 = \frac{u_1}{0.6}$
Multiply both sides by 0.6 to isolate u₁.
$u_1 = 45 \times 0.6 = 27$

Final answer:

u₁ = 27.

IB-style question — total distance of a bouncing ball

A ball is dropped from 12 m and rebounds to ½ of its height each bounce, forever. Find the total distance it travels.

Step by step:

It drops 12 m once; then each rebound is travelled up and back down. The rebound heights 6, 3, 1.5, … are geometric (u₁ = 6, r = ½) — sum them to infinity.
$S_\infty = \frac{6}{1 - \tfrac{1}{2}} = 12$
Total = the drop, plus twice the rebound sum.
$12 + 2(12) = 36$

Final answer:

36 m. (Shortcut: total = x(1 + r)/(1 − r); with r = ⅔ that's the classic δ = 5x.)

Important: S∞ = u₁/(1 − r) only works when |r| < 1. If |r| ≥ 1 there is no sum to infinity — the question wants a finite sum Sₙ (often the first 2m terms). Always check |r| before reaching for S∞.

Tap each card to reveal the answer.

Exam Tips

S∞ exists ONLY when |r| < 1; then S∞ = u₁/(1 − r). State the check first.
Find r as next ÷ current before doing anything else.
Given S∞, rearrange for u₁ or r — the denominator is 1 − r, never r.
Partial sums approach S∞: the gap is u₁rⁿ/(1 − r); set it below the tolerance and round up.
If |r| ≥ 1 there is no S∞ — give a finite sum Sₙ, simplifying with r²ᵐ = (r²)ᵐ.

IB Math AA SL — Infinite geometric series

Key concepts in Infinite geometric series

♾️ The sum to infinity

🔁 The exam variations

✏️ IB-style worked examples

IB-style question — find the sum to infinity

IB-style question — given S∞, find the first term

IB-style question — total distance of a bouncing ball

Exam Tips

What you'll learn in Topic 1.8

Study resources — 1.8 Infinite geometric series

Sum to infinity

Ready to study Infinite geometric series?

Ready to practice?

IB Math AA SL — Infinite geometric series

Key concepts in Infinite geometric series

♾️ The sum to infinity

🔁 The exam variations

✏️ IB-style worked examples

IB-style question — find the sum to infinity

IB-style question — given S∞, find the first term

IB-style question — total distance of a bouncing ball

Exam Tips

What you'll learn in Topic 1.8

Study resources — 1.8 Infinite geometric series

Sum to infinity

Ready to study Infinite geometric series?

Ready to practice?