Geometric Sequences

Geometric sequences are a powerful way to describe patterns that grow or shrink by the same factor each step. In this lesson, students, you will learn how to recognize a geometric sequence, use its key formulas, and connect it to real-life situations such as savings, population change, and depreciation 📈📉. These ideas are important in IB Mathematics: Applications and Interpretation HL because they help you model repeated change, make predictions, and interpret patterns using algebra and technology.

By the end of this lesson, you should be able to explain the meaning of terms such as common ratio and first term, identify whether a sequence is geometric, find any term in the sequence, and use the sum formula for a geometric series. You will also see how geometric sequences fit into the broader topic of Number and Algebra, where patterns, symbolic reasoning, and numerical modelling all work together.

What makes a sequence geometric?

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

If the first term is $u_1$, then the next terms are

$$u_2=u_1r, \quad u_3=u_1r^2, \quad u_4=u_1r^3$$

and so on.

For example, the sequence $3, 6, 12, 24, 48, \dots$ is geometric because each term is multiplied by $2$. Here, the first term is $u_1=3$ and the common ratio is $r=2$.

Not every pattern is geometric. In the sequence $4, 7, 10, 13, \dots$, the difference is constant, so it is arithmetic, not geometric. In a geometric sequence, the ratio between consecutive terms is constant:

$$\frac{u_2}{u_1}=\frac{u_3}{u_2}=\frac{u_4}{u_3}=r$$

This ratio test is one of the fastest ways to identify a geometric sequence.

A key idea is that geometric sequences can grow very quickly when $r>1$ or shrink rapidly when $0<r<1$. If $r$ is negative, the terms alternate in sign, such as $5, -10, 20, -40, \dots$ with $r=-2$. This alternating pattern is still geometric because the same multiplier is used each time.

The general term and how to use it

The $n$th term of a geometric sequence can be written using the formula

$$u_n=u_1r^{n-1}$$

This formula is extremely useful because it lets you find any term without listing all the earlier ones. The exponent $n-1$ appears because the first term already has no multiplication by $r$, the second term has one multiplication, the third has two, and so on.

Let’s use an example. Suppose $u_1=5$ and $r=3$. Then

$$u_n=5\cdot 3^{n-1}$$

To find the 6th term, substitute $n=6$:

$$u_6=5\cdot 3^5=5\cdot 243=1215$$

This formula also helps when a term number is known but the sequence is long. For example, in a population model, if a colony starts with $200$ bacteria and triples each hour, then after $t$ hours the number of bacteria is

$$N=200\cdot 3^t$$

This is not just a sequence; it is also a model of repeated multiplication over time. That makes geometric sequences especially important in financial modelling and science.

Sometimes you are given two terms and asked to find $r$. Suppose $u_1=8$ and $u_4=64$. Then

$$u_4=u_1r^3$$

$$64=8r^3$$

$$8=r^3$$

$$r=2$$

This method is common in IB-style questions because it uses algebraic reasoning rather than guessing.

Working with the sum of a geometric series

When the terms of a geometric sequence are added, the result is called a geometric series. The sum of the first $n$ terms is written as $S_n$.

If $r\ne 1$, the formula is

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

An equivalent form is

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

Both are correct. The choice depends on which is easier to use.

For example, find the sum of the first $5$ terms of the sequence $2, 6, 18, 54, 162, \dots$.

Here, $u_1=2$ and $r=3$. So

$$S_5=\frac{2(1-3^5)}{1-3}$$

$$S_5=\frac{2(1-243)}{-2}$$

$$S_5=242$$

You can also check by direct addition:

$$2+6+18+54+162=242$$

The formula saves time, especially when $n$ is large.

A useful real-world example is saving money with regular interest growth. If an investment grows by a factor each period, the total value after several periods can often be modelled using a geometric sequence. Another example is depreciation. If a car loses $15\%$ of its value each year, then it keeps $85\%$ of its value each year, so the ratio is $r=0.85$.

If the car is worth $18{,}000$ at the start, then after $4$ years its value is

$$V_4=18000(0.85)^4$$

This gives a realistic way to estimate value over time 🚗.

Recognizing geometric sequences in IB-style problems

IB Mathematics: Applications and Interpretation HL often tests whether you can interpret information from a context and choose the right model. To decide whether a situation is geometric, ask yourself: does the quantity change by adding the same amount, or by multiplying by the same factor?

If a phone battery loses $20\%$ each hour, then each hour it has $80\%$ of the previous amount, which means the model is geometric with $r=0.8$.

If a company’s revenue increases by $7\%$ each year, then each year’s value is multiplied by $1.07$. A model might look like

$$R_n=R_1(1.07)^{n-1}$$

This kind of reasoning is central to Number and Algebra because it links algebraic structure to numerical patterns.

Technology is often used to explore geometric sequences. A graphing calculator or spreadsheet can show the rapid growth or decay clearly. For example, if you enter the formula $u_n=100(1.12)^{n-1}$ into a table, you can see how values increase faster and faster. A spreadsheet can also be used to compare actual data with a geometric model and judge whether the model is suitable.

Sometimes the sequence is not exact because of rounding, measurement error, or changing conditions. In that case, the geometric model is still useful as an approximation, but you must interpret the results carefully. IB questions often reward clear communication of assumptions, such as assuming a constant percentage change.

Limits, behavior, and connections to algebra

Geometric sequences also connect to the idea of long-term behavior. When $0<r<1$, the terms get smaller and smaller. In fact,

$$\lim_{n\to\infty}u_1r^{n-1}=0$$

This happens because repeated multiplication by a number less than $1$ causes the sequence to decay toward zero.

For example, if

$$u_n=500(0.7)^{n-1}$$

the terms shrink quickly:

$$500, 350, 245, 171.5, \dots$$

This is useful in modelling radioactive decay, cooling, and depreciation.

If $r>1$, then the sequence grows without bound:

$$\lim_{n\to\infty}u_1r^{n-1}=\infty$$

for positive $u_1$. This reflects rapid growth in contexts like compound interest or population increase, although real systems often stop growing forever because of limits in the environment.

If $r<0$, the sequence alternates sign. For example,

$$u_n=4(-2)^{n-1}$$

gives $4, -8, 16, -32, \dots$ . The absolute values grow geometrically, but the signs switch each time. This can be important in pure mathematical problems.

Another algebraic connection is solving for unknown parameters. Suppose a geometric sequence has $u_1=6$ and $u_5=486$. Then

$$486=6r^4$$

$$81=r^4$$

$$r=3\quad \text{or} \quad r=-3$$

because both $3^4$ and $(-3)^4$ equal $81$. In some contexts, both answers are valid; in others, the situation may rule out a negative ratio. Always interpret the answer in context.

Conclusion

Geometric sequences describe repeated multiplication, making them one of the most important patterns in Number and Algebra. students, you should now be able to identify the common ratio $r$, use the general term formula $u_n=u_1r^{n-1}$, and find sums with the geometric series formula. You also saw how these ideas connect to modelling real situations such as savings, decay, and growth. In IB Mathematics: Applications and Interpretation HL, geometric sequences help you reason from a pattern, represent it algebraically, and interpret the results using technology and context. That combination of pattern, formula, and application is exactly what makes this topic so useful ✨.

Study Notes

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio $r$.
The common ratio is found by dividing consecutive terms: $r=\frac{u_{n+1}}{u_n}$, when $u_n\ne 0$.
The $n$th term of a geometric sequence is $u_n=u_1r^{n-1}$.
The sum of the first $n$ terms of a geometric series, for $r\ne 1$, is $S_n=\frac{u_1(1-r^n)}{1-r}$.
If $0<r<1$, the sequence decays toward $0$; if $r>1$, it grows rapidly.
If $r<0$, the terms alternate in sign.
Geometric sequences model repeated percentage change, such as interest, depreciation, and population growth.
Technology such as spreadsheets and graphing calculators helps test patterns and compare models with data.
In IB questions, always check whether the situation is better described by multiplication or addition.
Geometric sequences are a key part of Number and Algebra because they combine numerical patterns, algebraic formulas, and real-world modelling.