Geometric sequences and series

IB Mathematics: Applications and Interpretation SL· 1.3 Geometric sequences and series· 15 min read

1. Introduction to Geometric Sequences★★☆☆☆⏱ 5 min

📘 Definition

Geometric Sequence

u_n = u_1 r^{n-1}

A sequence where each consecutive term is the product of the previous term and a fixed constant called the common ratio $r$ , where $r \neq = 0$ .

Example:

Sequence $2, 6, 18, 54...$ has $u_{1} = 2$ and $r = 3$

Unlike arithmetic sequences that add a constant difference, geometric sequences multiply by a constant ratio. The common ratio can be positive, negative, or a fraction less than 1.

📐 Worked Example

Determine if the sequence $4, - 2, 1, - 0.5, 0.25...$ is geometric. If it is, find the common ratio and the 6th term.

1
Calculate the ratio between each pair of consecutive terms:
2
$\frac{-2}{4} = -0.5, \quad \frac{1}{-2} = -0.5, \quad \frac{-0.5}{1} = -0.5$
3
The ratio is constant, so this is a geometric sequence with common ratio $r = - 0.5$ and first term $u_{1} = 4$ .
4
Use the nth term formula to find $u_{6}$ :
5
$u_6 = u_1 r^{6-1} = 4(-0.5)^5 = 4 \times -\frac{1}{32} = -0.125$

Exam tip:

Always check the ratio between at least three consecutive pairs to confirm a sequence is geometric, don't just check one.

2. Finite Geometric Series★★★☆☆⏱ 6 min

📘 Definition

Finite Geometric Series

S_n

The sum of the first $n$ terms of a geometric sequence, calculated with the formula:

S_n = \frac{u_1 (1 - r^n)}{1 - r} = \frac{u_1 (r^n - 1)}{r - 1}, \quad r \neq 1

If $r = 1$ , the sequence is constant (all terms equal to $u_{1}$ ), so the sum simplifies to $S_{n} = n u_{1}$ . This formula is used for adding up a fixed number of terms, common in annuity and investment problems.

📐 Worked Example

Find the sum of the first 10 terms of the geometric sequence $3, 6, 12, 24...$

1
Identify the values: $u_{1} = 3$ , $r = 2$ , $n = 10$
2
Substitute into the sum formula:
3
$S_{10} = \frac{3(2^{10} - 1)}{2 - 1} = 3(1024 - 1) = 3 \times 1023 = 3069$

3. Infinite Geometric Series and Convergence★★★☆☆⏱ 5 min

📘 Definition

Convergent Infinite Geometric Series

S_\infty

An infinite geometric series approaches a finite sum only when the absolute value of the common ratio is less than 1 ( $∣ r ∣ < 1$ ). If $∣ r ∣ \geq 1$ , the series diverges and has no finite sum.

Example:

The series $1 + 0.5 + 0.25 + ...$ converges, $1 + 2 + 4 + ...$ diverges

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

📐 Worked Example

Find the sum to infinity of the series $12 + 8 + \frac{16}{3} + \frac{32}{9} + ...$

1
Calculate the common ratio:
2
$r = \frac{8}{12} = \frac{2}{3}$
3
Check convergence condition: $∣ r ∣ = \frac{2}{3} < 1$ , so the series converges.
4
Substitute into the sum to infinity formula:
5
$S_\infty = \frac{12}{1 - \frac{2}{3}} = \frac{12}{\frac{1}{3}} = 36$

4. Real-World Applications★★★★☆⏱ 7 min

Geometric sequences are the standard model for exponential growth and decay in real-world contexts including compound interest, population change, radioactive decay, and recurring investments.

📐 Worked Example

Find the total value of annual investments of $1000 made at the start of each year for 10 years, with a constant 5% annual compound interest rate.

1
The first deposit earns interest for 10 years, the second for 9 years, and the last deposit earns interest for 1 year. The total value is a geometric series:
2
$u_{1} = 1000 \times 1.05$ , $r = 1.05$ , $n = 10$
3
Substitute into the finite sum formula:
4
$S_{10} = \frac{1000 \times 1.05 (1.05^{10} - 1)}{1.05 - 1} \approx 13206.74$

5. Common Pitfalls

Wrong move:

Using the nth term formula $u_{n} = u_{1} r^{n}$ instead of $u_{n} = u_{1} r^{n - 1}$

Why:

This causes an off-by-one exponent error that gives the wrong term, especially for early terms in the sequence

Correct move:

Test the formula with $n = 1$ : $u_{1} = u_{1} r^{0} = u_{1}$ , so the exponent must be $n - 1$

Wrong move:

Using the sum to infinity formula without checking $∣ r ∣ < 1$

Why:

Examiners explicitly award a mark for this check, and the formula is invalid for divergent series

Correct move:

Always calculate $∣ r ∣$ and state that it is less than 1 before applying the $S_{\infty}$ formula

Wrong move:

Getting a negative sum for a series with $r > 1$

Why:

Mixing up the order of terms in the numerator of the sum formula

Correct move:

Use $S_{n} = \frac{u _{1} ( r ^{n} - 1 )}{r - 1}$ for $r > 1$ , which always gives a positive sum matching all positive terms

Wrong move:

Counting the wrong number of compounding periods for investments

Why:

Confusing whether the deposit is made at the start or end of the year

Correct move:

Draw a 2-year test case: a deposit at the start of year 1 earns interest for 1 year by the end of year 1, so adjust your exponent accordingly

6. Quick Reference Cheatsheet

Concept	Formula	Required Condition
nth term	$u_{n} = u_{1} r^{n - 1}$	Always true
Sum of first $n$ terms	$S_{n} = \frac{u _{1} ( 1 - r ^{n} )}{1 - r}$	$r \neq = 1$
Sum to infinity	$S_{\infty} = \frac{u _{1}}{1 - r}$	$∣ r ∣ < 1$
Single compound investment	$V_{n} = P (1 + \frac{R}{100})^{n}$	R = annual % rate
Annual annuity total	$S_{n} = \frac{D ( 1 + r ) [( 1 + r ) ^{n} - 1 ]}{r}$	D = annual deposit

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Compound interest problem
2023 · 1
Population decay calculation
2024 · 2
Annuity total value problem

Going deeper

syllabusIB AI SL Official Syllabus Topic 1Official topic guidance

What's Next

Geometric sequences and series form the foundation for modeling exponential growth and decay, which you will explore further in functions and statistical topics in IB AI SL. This topic is also the basis for financial mathematics applications that make up a significant portion of exam marks, appearing regularly in both Paper 1 and Paper 2. Mastery of convergence and common ratio properties also supports understanding of recursive sequences and probability distributions later in the course.