Geometric Sequences and Series

Welcome, students 👋 Today you will learn one of the most useful patterns in algebra: geometric sequences and geometric series. These ideas show up in finance, science, computer science, and anywhere something changes by the same factor each time. By the end of this lesson, you should be able to recognize a geometric pattern, write its terms and formulas, and calculate sums of its terms with confidence.

Objectives

Explain the main ideas and terminology behind geometric sequences and series.
Apply IB Mathematics Analysis and Approaches SL methods to find terms and sums.
Connect geometric sequences and series to the broader topic of Number and Algebra.
Summarize why these patterns matter in mathematical modeling.
Use examples to justify your answers and spot common structures.

A quick hook: imagine a bacteria culture that doubles every hour, or money in a savings account that grows by a fixed percentage each year. In both cases, the quantity is multiplied by the same number repeatedly. That repeated multiplication is exactly what makes a pattern geometric 📈.

What is a geometric sequence?

A sequence is an ordered list of numbers. In a geometric sequence, each term is found by multiplying the previous term by the same constant number. That constant is called the common ratio, written as $r$.

If the first term is $a$, then the terms of the sequence are often written as

$$a,\ ar,\ ar^2,\ ar^3,\dots$$

Here, $a$ is the first term, and $r$ is the common ratio. For a geometric sequence, the ratio between consecutive terms is always the same:

$$\frac{u_{n+1}}{u_n}=r$$

for terms where $u_n \neq 0$.

Example 1

Consider the sequence

$$3,\ 6,\ 12,\ 24,\dots$$

Each term is multiplied by $2$, so the common ratio is $r=2$. This is a geometric sequence with first term $a=3$.

The next terms would be:

$$48,\ 96,\ 192$$

because each time we multiply by $2$.

Example 2

Consider

$$81,\ 27,\ 9,\ 3,\dots$$

The common ratio is

$$r=\frac{27}{81}=\frac{1}{3}$$

So this sequence is geometric with $a=81$ and $r=\frac{1}{3}$. A ratio less than $1$ makes the numbers shrink over time, which is common in depreciation or repeated decay.

Important terminology

Term: one number in the sequence.
First term: the starting value, often $a$ or $u_1$.
Common ratio: the constant multiplier $r$.
Geometric: patterns based on multiplication rather than addition.

A helpful contrast is with an arithmetic sequence, where terms change by adding the same amount. In a geometric sequence, the change is multiplicative, not additive.

The nth term of a geometric sequence

To find any term directly, use the general formula for the $n$th term:

$$u_n=ar^{n-1}$$

This formula works because the first term is $u_1=a$, the second term is $u_2=ar$, and each new term multiplies by $r$ again.

Why the formula makes sense

For $n=1$,

$$u_1=ar^{0}=a$$

For $n=2$,

$$u_2=ar^{1}=ar$$

For $n=3$,

$$u_3=ar^{2}$$

and so on.

Example 3

A geometric sequence has $a=5$ and $r=3$.

Find the $6$th term.

Using

$$u_n=ar^{n-1}$$

we get

$$u_6=5\cdot 3^{5}$$

$$u_6=5\cdot 243=1215$$

So the sixth term is $1215$.

Example 4

A sequence has first term $a=64$ and common ratio $r=\frac{1}{2}$.

Find the $4$th term.

$$u_4=64\left(\frac{1}{2}\right)^3=64\cdot \frac{1}{8}=8$$

This pattern is useful when a quantity halves repeatedly, such as the amount of some substance remaining after several time periods.

Finding the common ratio from terms

If two consecutive terms are given, divide the later term by the earlier term.

For example, if the sequence is

$$2,\ 10,\ 50,\dots$$

then

$$r=\frac{10}{2}=5$$

and also

$$r=\frac{50}{10}=5$$

The fact that the same ratio appears each time confirms it is geometric.

Geometric series and the sum of terms

A series is the sum of terms in a sequence. So if a geometric sequence is

$$a,\ ar,\ ar^2,\ ar^3,\dots$$

a geometric series looks like

$$a+ar+ar^2+ar^3+\dots$$

Sometimes you need the sum of only the first $n$ terms. This is called the partial sum.

The formula for the sum of the first $n$ terms is

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

for $r\neq 1$.

An equivalent form is

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

Both are correct. Use whichever is more convenient.

Example 5

Find the sum of the first $4$ terms of

$$2,\ 6,\ 18,\ 54,\dots$$

Here $a=2$ and $r=3$. Use

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

$$S_4=\frac{2(1-3^4)}{1-3}$$

$$S_4=\frac{2(1-81)}{-2}$$

$$S_4=\frac{2(-80)}{-2}=80$$

Check by direct addition:

$$2+6+18+54=80$$

The formula matches the direct sum ✅.

Example 6

Find the sum of the first $5$ terms of

$$160,\ 80,\ 40,\ 20,\ 10,\dots$$

Here $a=160$ and $r=\frac{1}{2}$.

$$S_5=\frac{160\left(1-\left(\frac{1}{2}\right)^5\right)}{1-\frac{1}{2}}$$

$$S_5=\frac{160\left(1-\frac{1}{32}\right)}{\frac{1}{2}}$$

$$S_5=\frac{160\cdot \frac{31}{32}}{\frac{1}{2}}$$

$$S_5=310$$

This kind of calculation can model the total amount received after repeated smaller payments or the total distance traveled if each step gets shorter by the same factor.

Special cases and interpretation

Geometric sequences can behave very differently depending on the value of $r$.

1. If $r>1$

The terms grow quickly. Example:

$$4,\ 12,\ 36,\ 108,\dots$$

This is exponential growth in sequence form.

2. If $0<r<1$

The terms get smaller and approach $0$.

Example:

$$100,\ 50,\ 25,\ 12.5,\dots$$

This models decay or reduction.

3. If $r<0$

The terms change sign each step.

Example:

$$5,\ -10,\ 20,\ -40,\dots$$

This still counts as geometric because the ratio is constant:

$$\frac{-10}{5}=-2$$

and

$$\frac{20}{-10}=-2$$

4. If $r=1$

Every term is the same.

$$7,\ 7,\ 7,\ 7,\dots$$

This is geometric, but it is a very simple case.

5. If $r=0$

The sequence becomes

$$a,\ 0,\ 0,\ 0,\dots$$

This is also geometric once the ratio is defined from the terms.

The formula for $S_n$ requires special care when $r=1$, because then the denominator becomes $0$. In that case, the sum is simply

$$S_n=na$$

since all $n$ terms are equal.

Why this matters in IB Mathematics Analysis and Approaches SL

Geometric sequences and series are part of Number and Algebra because they combine pattern recognition, symbolic manipulation, and reasoning about growth. They connect directly to exponentials, since

$$ar^{n-1}$$

is an exponential expression in $n$ when $r$ is fixed.

They also connect to real-world modeling. For example:

compound interest uses repeated multiplication,
population growth can sometimes be modeled approximately with a constant ratio,
radioactive decay uses a ratio less than $1$,
recursive patterns in computer algorithms can follow geometric structure.

In IB mathematics, you are expected not just to memorize formulas, but to understand when they apply and to explain your working clearly. That means checking whether a sequence has a constant ratio, identifying $a$ and $r$, and choosing the right formula for the question.

Example 7: context problem

A new phone costs $900$. Its value decreases by $15\%$ each year.

The yearly multiplier is

$$r=0.85$$

So after $n$ years, the value is modeled by

$$V_n=900(0.85)^{n-1}$$

If you want the value after $4$ years, use

$$V_4=900(0.85)^3$$

This gives a realistic example of a geometric pattern in everyday life 📱.

Conclusion

Geometric sequences and series are built on one powerful idea: repeated multiplication by a constant ratio. The sequence formula

$$u_n=ar^{n-1}$$

lets you find any term, while the sum formula

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

lets you add many terms efficiently. These tools are essential in IB Mathematics Analysis and Approaches SL because they strengthen your ability to recognize patterns, manipulate symbols, and model real situations accurately. students, if you can identify $a$ and $r$, you already have the key to solving many geometric sequence and series problems.

Study Notes

A geometric sequence multiplies by the same constant each time.
The constant multiplier is the common ratio, $r$.
The first term is $a$.
The $n$th term is $u_n=ar^{n-1}$.
A geometric series is the sum of terms from a geometric sequence.
The sum of the first $n$ terms is $S_n=\frac{a(1-r^n)}{1-r}$ for $r\neq 1$.
If $r=1$, then $S_n=na$.
To check if a sequence is geometric, divide consecutive terms and see whether the ratio stays constant.
If $r>1$, terms grow; if $0<r<1$, terms shrink; if $r<0$, terms alternate signs.
Geometric sequences connect to exponentials, growth, decay, and financial modeling.