Geometric Sequences

Welcome, students! 📘 In this lesson, you will learn how geometric sequences work, why they appear in real life, and how to use them in IB Mathematics: Applications and Interpretation SL. A geometric sequence is one of the most important patterns in Number and Algebra because it shows repeated multiplication rather than repeated addition. That simple idea helps model population growth, money in bank accounts, virus spread, and even the shrinking of a bouncing ball. By the end of this lesson, you should be able to identify geometric sequences, find the common ratio, write a sequence rule, and use the formula for the $n$th term. You will also see how technology can support checking patterns and making predictions. 🚀

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 number each time. That fixed number is called the common ratio, usually written as $r$.

For example, the sequence $3, 6, 12, 24, 48, \dots$ is geometric because each term is multiplied by $2$ to get the next term. Here, the common ratio is $r=2$.

Another example is $81, 27, 9, 3, 1, \dots$ . This is geometric too, because each term is multiplied by $\frac{1}{3}$. So the common ratio is $r=\frac{1}{3}$.

A geometric sequence can grow very fast when $|r|>1$, or shrink toward $0$ when $0<|r|<1$. If $r$ is negative, the terms alternate between positive and negative values. For example, $5, -10, 20, -40, \dots$ has common ratio $r=-2$.

It is important to distinguish a geometric sequence from an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant. This difference is a key idea in Number and Algebra because it helps you decide which model fits a situation.

Finding the common ratio

To check whether a sequence is geometric, divide each term by the term before it. If the quotient is always the same, the sequence is geometric.

For the sequence $4, 12, 36, 108, \dots$:

$\frac{12}{4}=3$
$\frac{36}{12}=3$
$\frac{108}{36}=3$

Since the ratio is constant, the sequence is geometric and $r=3$.

For a sequence like $2, 6, 18, 54, \dots$, the ratio is also constant because $\frac{6}{2}=3$, $\frac{18}{6}=3$, and $\frac{54}{18}=3$. This means each term is three times the previous term. In real life, this could represent money growing by a factor of $3$, although that would be unusual in a bank account. More commonly, a geometric sequence models repeated percentage change, such as growth by $5\%$ each period, where the multiplier is $1.05$.

Be careful: if the sequence contains zero, you cannot divide by zero to find a ratio. Also, if the sequence is not perfectly constant in ratio, it is not geometric.

The formula for the $n$th term

Once you know the first term and the common ratio, you can write a general rule for any term in the sequence. If the first term is $u_1=a$ and the common ratio is $r$, then the $n$th term is

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

This formula is one of the most useful results in this topic.

Why does it work? The first term is $a$. The second term is $ar$. The third term is $ar^2$. The fourth term is $ar^3$. Each time, the power of $r$ increases by $1$, so by the time you reach the $n$th term, the exponent is $n-1$.

Example: Find the $8$th term of the geometric sequence $5, 15, 45, \dots$.

Here, $a=5$ and $r=3$. Use the formula:

$$u_8=5\cdot 3^{8-1}=5\cdot 3^7$$

Since $3^7=2187$,

$$u_8=10935$$

So the $8$th term is $10935$.

Another example: For the sequence $64, 32, 16, 8, \dots$, we have $a=64$ and $r=\frac{1}{2}$. Then

$$u_n=64\left(\frac{1}{2}\right)^{n-1}$$

This sequence gets smaller each time because the ratio is between $0$ and $1$.

Using geometric sequences in real life

Geometric sequences are very useful for financial models and numerical modelling. Many situations involve multiplying by the same factor repeatedly.

Money and compound growth

Suppose you invest $1000$ in an account that grows by $4\%$ each year. After one year, the amount becomes $1000\times 1.04$. After two years, it becomes $1000\times 1.04^2$. After $n$ years, the model is

$$A_n=1000(1.04)^n$$

This is a geometric sequence because each year the amount is multiplied by the same ratio $r=1.04$.

If you want the amount after $5$ years, calculate

$$A_5=1000(1.04)^5$$

Technology such as a graphing calculator or spreadsheet can help you evaluate this quickly and see how the amount increases over time. In IB Mathematics: Applications and Interpretation SL, using technology to interpret patterns is an important skill. 📱

Depreciation and shrinking models

Geometric sequences also model decrease. A car that loses $15\%$ of its value each year keeps $85\%$ of its value each year. If the value starts at $20000$, then the model after $n$ years is

$$V_n=20000(0.85)^n$$

Because the ratio is $0.85$, the sequence decreases but does not usually reach zero exactly. This type of model is useful when studying depreciation of vehicles, equipment, or electronics.

Population and repeated change

A population that grows by a fixed percentage each time period can also be represented by a geometric sequence. For example, if a town has $5000$ people and the population grows by $2\%$ per year, then after $n$ years,

$$P_n=5000(1.02)^n$$

This is an example of exponential growth written as a sequence. The key link is that each term is created by multiplying the previous one by the same ratio.

Working with terms and solving problems

In IB-style questions, you may be asked to find an unknown term, the common ratio, or the first term.

If you know two consecutive terms, you can find the ratio by dividing. For example, if $u_4=54$ and $u_5=81$, then

$$r=\frac{u_5}{u_4}=\frac{81}{54}=\frac{3}{2}$$

If you know the first term and the ratio, you can find any term using $u_n=ar^{n-1}$. If you know a term and the ratio, you can work backward or set up an equation.

Example: A geometric sequence has $u_1=2$ and $r=4$. Find $u_6$.

Use the formula:

$$u_6=2\cdot 4^{5}$$

Since $4^5=1024$,

$$u_6=2048$$

Sometimes a question may ask you to identify whether a real situation is geometric. If a quantity changes by the same percentage each time, it is geometric. If it changes by the same amount each time, it is arithmetic. That distinction is a common exam focus and helps you choose the correct model.

You may also need to interpret a ratio. If $r>1$, the sequence grows. If $0<r<1$, it decays. If $r<0$, the sign changes each term. If $r=1$, the sequence is constant. These cases are useful for quick reasoning.

Technology-supported interpretation

Technology can help you explore geometric sequences more deeply. A spreadsheet can generate terms quickly using formulas. A graphing calculator can display values and help you spot patterns. Plotting term number against term value can show growth or decay, although the graph will not be a straight line like an arithmetic sequence.

For example, entering $u_n=3(1.5)^{n-1}$ into technology lets you compare terms, see rapid growth, and predict future values. You can also use technology to check whether your answer makes sense. If the ratio is greater than $1$, the terms should increase. If your calculated terms decrease instead, you may have made an error.

In IB Mathematics: Applications and Interpretation SL, technology is not just for arithmetic. It is used to interpret results, validate models, and connect mathematics to real situations. That means you should understand the sequence concept first, then use technology to support your reasoning.

Conclusion

Geometric sequences are a major part of Number and Algebra because they show how repeated multiplication creates patterns. The central ideas are the first term $a$, the common ratio $r$, and the formula $u_n=ar^{n-1}$. These sequences model growth, decay, and repeated percentage change in many real-world situations, including finance, population change, and depreciation. They also connect strongly to technology-supported interpretation, which is an important feature of IB Mathematics: Applications and Interpretation SL. If you can recognize a geometric sequence, find its ratio, and use the $n$th-term formula, you have a strong foundation for more advanced modelling tasks. ✅

Study Notes

A geometric sequence is a sequence where each term is found by multiplying the previous term by the same number.
The fixed multiplier is called the common ratio, written as $r$.
To check if a sequence is geometric, divide each term by the one before it.
The formula for the $n$th term is $u_n=ar^{n-1}$, where $a$ is the first term.
If $r>1$, the sequence grows; if $0<r<1$, it decays; if $r<0$, the signs alternate.
Geometric sequences model repeated percentage growth and decay, such as compound interest and depreciation.
A change by the same amount each time is arithmetic, not geometric.
Technology such as spreadsheets and graphing calculators can help generate terms and check models.
In IB Mathematics: Applications and Interpretation SL, geometric sequences are part of numerical modelling and algebraic representation.
Always interpret the ratio carefully, because it tells you how the sequence changes over time.