Geometric sequences and series

Q: What is a geometric sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value called the common ratio ($r$).

Q: How do you calculate the common ratio ($r$) of a geometric sequence?

The common ratio is found by dividing any term by the term immediately preceding it: $r = \frac{u_{n+1}}{u_n}$.

Q: What is a geometric series?

A geometric series is the sum of the terms of a geometric sequence, denoted by $S_n$.

Q: What are the formulas for the sum of the first $n$ terms ($S_n$) of a geometric sequence?

$S_n = \frac{u_1(r^n - 1)}{r - 1}$ or $S_n = \frac{u_1(1 - r^n)}{1 - r}$

Q: What does Sigma notation ($\Sigma$) represent in geometric sequences?

Sigma notation is a shorthand way to write the sum of a sequence, showing the starting index, the ending index, and the general term formula.

Q: Under what condition does an infinite geometric series converge? (HL Only)?

An infinite geometric series converges only if the absolute value of the common ratio is less than 1 ($|r| < 1$).

Q: What is the formula for the sum to infinity of a convergent geometric series? (HL Only)?

$S_\infty = \frac{u_1}{1 - r}$

Q: How is the common ratio ($r$) determined for percentage growth or decay?

For growth of $x\%$, $r = 1 + \frac{x}{100}$; for decay of $x\%$, $r = 1 - \frac{x}{100}$.

Mathematics - Applications and Interpretation · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: 7 Sept 2026

Introduction to Geometric Sequences and Series

In the previous chapter, we looked at arithmetic sequences, where we added the same number each time. In this chapter, we explore geometric sequences, where each term is found by multiplying the previous term by a constant value. Whether it is a viral video doubling its views every hour or a bouncing ball losing a fraction of its height each time it hits the ground, geometric sequences are everywhere in the real world.

If you enjoy seeing how things grow or shrink exponentially, this is the topic for you! Don't worry if the formulas look a bit intimidating at first; we will break them down step-by-step.

1. What is a Geometric Sequence?

A geometric sequence is a list of numbers where the ratio between any two consecutive terms is constant. We call this constant the common ratio, denoted by the letter $r$.

The Core Components:
$u_1$: The first term of the sequence.
$r$: The common ratio.
$n$: The position of the term (e.g., $n=5$ for the 5th term).
$u_n$: The value of the $n$th term.

How to find the common ratio ($r$):
To find $r$, simply divide any term by the term that comes immediately before it:
$r = \frac{u_2}{u_1} = \frac{u_3}{u_2} = \frac{u_{n+1}}{u_n}$

Example: In the sequence $3, 6, 12, 24, ...$
The first term $u_1 = 3$.
The common ratio $r = \frac{6}{3} = 2$.

Quick Review:
If $r > 1$, the numbers are getting larger (growth).
If $0 < r < 1$, the numbers are getting smaller (decay).
If $r$ is negative, the terms will alternate between positive and negative values!

2. Finding the $n$th Term

If you want to find the 100th term of a sequence, you don't want to multiply by $r$ a hundred times! Instead, we use the general formula found in your IB Formula Booklet:

The Formula:
$u_n = u_1 \cdot r^{n-1}$

Why $n-1$? Think about it this way: to get to the 2nd term, you multiply by $r$ once. To get to the 3rd term, you multiply by $r$ twice. So, to get to the $n$th term, you always multiply the first term by $r$ one less time than the position you are looking for.

Step-by-Step Example:
Find the 8th term of the sequence $10, 5, 2.5, ...$
1. Identify $u_1$: $u_1 = 10$.
2. Identify $r$: $r = \frac{5}{10} = 0.5$.
3. Use the formula: $u_8 = 10 \cdot (0.5)^{8-1}$.
4. Calculate: $u_8 = 10 \cdot (0.5)^7 = 0.078125$.

Key Takeaway: Always identify your "ingredients" ($u_1$, $r$, and $n$) before plugging them into the formula.

3. Geometric Series (Summing the Terms)

A series is what we get when we add the terms of a sequence together. The sum of the first $n$ terms is denoted by $S_n$.

The Formulas:
$S_n = \frac{u_1(r^n - 1)}{r - 1}$ or $S_n = \frac{u_1(1 - r^n)}{1 - r}$

Note: You can use either formula, but usually, we use the first one if $r > 1$ and the second one if $r < 1$ to keep the numbers positive and easy to handle.

Using Technology:
Since this is an Applications and Interpretation course, your Graphic Display Calculator (GDC) is your best friend. You can often solve these problems using the "Summation" tool or the "Numerical Solver" on your calculator. Make sure you practice entering these formulas correctly!

4. Sigma Notation

Sigma notation is just a mathematical shorthand for writing a sum. It uses the Greek letter $\Sigma$ (Sigma).

What it looks like:
$\sum_{k=1}^{n} u_1 \cdot r^{k-1}$

How to read it:
1. The bottom number ($k=1$) tells you where to start.
2. The top number ($n$) tells you where to stop.
3. The expression to the right is the formula for the sequence.

Did you know? You can enter sigma notation directly into most GDCs to find the sum automatically without having to manually use the $S_n$ formula!

5. Infinite Geometric Series (HL Only)

Imagine you have a cake and you eat half of it. Then you eat half of what is left. Then half of that. If you keep doing this forever, will you ever eat a total of two cakes? No! You will eventually just finish that one original cake.

This is the concept of a convergent series. If the common ratio $r$ is between $-1$ and $1$ (written as $|r| < 1$), the sum of the sequence will get closer and closer to a specific number as $n$ goes to infinity.

The Formula for the Sum to Infinity:
$S_\infty = \frac{u_1}{1 - r}$ (only if $|r| < 1$)

Common Mistake: Students often try to use this formula when $r$ is larger than 1. If $r = 2$, the sequence just keeps getting bigger and bigger ($2, 4, 8, 16...$), so the sum is infinity. The formula only works when the terms are "dying out" toward zero.

6. Applications and Modeling

Geometric sequences are rarely just abstract numbers in this course; they usually represent real-life situations:

Population Growth: If a population grows by $5\%$ each year, the common ratio is $r = 1.05$.
Depreciation: If a car loses $15\%$ of its value each year, the common ratio is $r = 0.85$. (Cross-reference: See SL 1.4 for more on Financial Mathematics.)
Predictions: We use these models to predict future values. However, keep in mind that in real life, models aren't always perfect. A population might stop growing due to limited resources.

Problem-Solving Tip: If a question says "increases by $x\%$", your ratio is $r = 1 + \frac{x}{100}$. If it says "decreases by $x\%$", your ratio is $r = 1 - \frac{x}{100}$.

Summary Checklist

1. Identify the type: Is it geometric? (Check if there is a common multiplier $r$).
2. Pick your tool: Use $u_n$ for a specific term and $S_n$ for a total sum.
3. Check your $r$: If you are summing to infinity (HL), ensure $|r| < 1$.
4. Calculator check: Ensure you are comfortable using your GDC for exponents and large sums.
5. Units: In application questions, always include the units (e.g., \$, people, cm) in your final answer.

Don't worry if this seems tricky at first! Geometric growth is much faster than the arithmetic growth we are used to, so the numbers can get very large or very small very quickly. Trust the formulas and your calculator!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a geometric sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value called the common ratio ($r$).

How do you calculate the common ratio ($r$) of a geometric sequence?

The common ratio is found by dividing any term by the term immediately preceding it: $r = \frac{u_{n+1}}{u_n}$.

What is the formula for the $n$th term of a geometric sequence?

$u_n = u_1 \cdot r^{n-1}$

What is a geometric series?

A geometric series is the sum of the terms of a geometric sequence, denoted by $S_n$.

What are the formulas for the sum of the first $n$ terms ($S_n$) of a geometric sequence?

$S_n = \frac{u_1(r^n - 1)}{r - 1}$ or $S_n = \frac{u_1(1 - r^n)}{1 - r}$

What does Sigma notation ($\Sigma$) represent in geometric sequences?

Sigma notation is a shorthand way to write the sum of a sequence, showing the starting index, the ending index, and the general term formula.

Under what condition does an infinite geometric series converge? (HL Only)

An infinite geometric series converges only if the absolute value of the common ratio is less than 1 ($|r| < 1$).

What is the formula for the sum to infinity of a convergent geometric series? (HL Only)

$S_\infty = \frac{u_1}{1 - r}$

How is the common ratio ($r$) determined for percentage growth or decay?

For growth of $x\%$, $r = 1 + \frac{x}{100}$; for decay of $x\%$, $r = 1 - \frac{x}{100}$.

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