Change in Arithmetic and Geometric Sequences 📈

students, in this lesson you will learn how sequences change over time and why that matters in AP Precalculus. The big idea is simple: some patterns grow by adding the same amount each step, while others grow by multiplying by the same factor each step. Those two ideas are called arithmetic sequences and geometric sequences. You will see how to identify them, describe their change, and connect them to exponential and logarithmic functions.

Introduction: Why sequence change matters

Imagine a savings plan, a staircase, or a phone data plan. In each situation, the amount can change in a predictable pattern. If a worker gets a raise of $500$ dollars every year, that is a constant additive change. If money in an account grows by $5\%$ each year, that is a constant multiplicative change. These are the two main sequence types in this lesson.

By the end of this lesson, students, you should be able to:

explain the meaning of arithmetic and geometric sequences,
describe how their terms change,
use formulas to find missing terms,
connect geometric sequences to exponential functions, and
recognize when logarithms can help solve for the number of steps in a pattern.

Arithmetic sequences: constant additive change ➕

An arithmetic sequence is a list of numbers where each term changes by the same amount. That fixed amount is called the common difference, written as $d$. If the terms are $a_1, a_2, a_3, \dots$, then an arithmetic sequence satisfies

$$a_{n}=a_{n-1}+d.$$

This means each term is found by adding $d$ to the previous term. For example, the sequence $3, 7, 11, 15, \dots$ is arithmetic because the common difference is $4$.

You can also write the $n$th term with the explicit formula

$$a_n=a_1+(n-1)d.$$

This formula is useful because it lets you jump directly to any term without listing all the earlier ones.

Example 1: Finding an arithmetic term

Suppose a school club starts with $12$ members and gains $3$ members each month. The sequence is

$$12, 15, 18, 21, \dots$$

Here, $a_1=12$ and $d=3$. To find the $10$th term, use

$$a_{10}=12+(10-1)(3)=12+27=39.$$

So the club would have $39$ members in month $10$ if the pattern continues.

Change in arithmetic sequences

The key idea of change here is the difference between consecutive terms. For an arithmetic sequence, the change is always the same:

$$a_n-a_{n-1}=d.$$

This kind of change is linear in spirit because the amount added does not depend on how large the terms already are. That makes arithmetic sequences very different from exponential patterns, where growth depends on the current value.

A real-world example is saving a fixed amount of money each week. If you save $20$ dollars every week, your total after each week increases by exactly $20$. The pattern is predictable and steady ✅

Geometric sequences: constant multiplicative change ✨

A geometric sequence is a list of numbers where each term is found by multiplying by the same number. That fixed multiplier is called the common ratio, written as $r$. If the terms are $a_1, a_2, a_3, \dots$, then a geometric sequence satisfies

$$a_n=a_{n-1}r.$$

The explicit formula is

$$a_n=a_1r^{n-1}.$$

This formula looks a lot like an exponential function because the variable $n$ is in the exponent.

Example 2: Finding a geometric term

Suppose bacteria in a lab triple each hour. If the starting population is $50$, the sequence is

$$50, 150, 450, 1350, \dots$$

Here, $a_1=50$ and $r=3$. The $6$th term is

$$a_6=50(3)^{6-1}=50(3^5)=50(243)=12150.$$

So after $6$ hours, the model gives $12150$ bacteria.

Change in geometric sequences

The change in a geometric sequence is not constant in difference; instead, the ratio stays constant:

$$\frac{a_n}{a_{n-1}}=r.$$

That means each term grows by the same percentage or multiplies by the same factor. This is the same idea behind many exponential situations, such as population growth, compound interest, and inflation.

For example, if a concert ticket price increases by $8\%$ each year, the price is multiplied by $1.08$ each year. That is geometric growth. The actual increase gets larger over time because the increase depends on the current value 🎵

Connecting sequences to exponential and logarithmic functions 🔗

Geometric sequences are closely connected to exponential functions. A geometric sequence is discrete, meaning it is defined at whole-number terms like $a_1, a_2, a_3, \dots$. An exponential function is continuous, meaning it can be evaluated for any real input.

A geometric sequence with first term $a_1$ and ratio $r$ can be written as

$$a_n=a_1r^{n-1}.$$

This is very similar to an exponential function of the form

$$f(x)=ab^x.$$

If you let $x=n-1$, the sequence formula becomes an exponential expression. This is why exponential functions are used to model repeated multiplicative change.

Logarithms enter the picture when you need to solve for the exponent. For example, suppose a quantity follows

$$a_n=100(1.2)^{n-1}$$

and you want to know when it first exceeds $200$. You solve

$$100(1.2)^{n-1}>200.$$

Divide by $100$:

$$ (1.2)^{n-1}>2.$$

To find $n$, take logarithms:

$$n-1>\log_{1.2}(2).$$

This shows how logarithms help reverse exponential growth. In AP Precalculus, this connection is important because many questions ask you to compare growth, solve for time, or interpret when a quantity reaches a target value.

How to tell arithmetic and geometric sequences apart 🧠

A fast way to identify the sequence type is to check consecutive differences and ratios.

If the difference $a_n-a_{n-1}$ is constant, the sequence is arithmetic.
If the ratio $\frac{a_n}{a_{n-1}}$ is constant, the sequence is geometric.

Example 3: Decide the type

Consider the sequence $2, 6, 18, 54, \dots$

Check the differences:

$$6-2=4,\quad 18-6=12,\quad 54-18=36.$$

These are not constant. Now check the ratios:

$$\frac{6}{2}=3,\quad \frac{18}{6}=3,\quad \frac{54}{18}=3.$$

Because the ratio is constant, this is geometric with $r=3$.

Now consider $9, 14, 19, 24, \dots$

The differences are

$$14-9=5,\quad 19-14=5,\quad 24-19=5.$$

This is arithmetic with $d=5$.

What “change” means in AP Precalculus

students, “change” in this topic means more than just increasing or decreasing. It means identifying the rule that describes how one term becomes the next. In arithmetic sequences, the change is additive. In geometric sequences, the change is multiplicative. These two forms of change support much of the reasoning in exponential modeling.

If a sequence is decreasing, the same ideas still work. An arithmetic sequence can decrease if $d<0$. A geometric sequence can decrease if $0<r<1$ or if $r<0$ causes alternating signs. For example, the sequence $81, 27, 9, 3, \dots$ is geometric with $r=\frac{1}{3}$.

AP-style reasoning and applications 📝

AP Precalculus often asks you to interpret, compare, and extend patterns. Here are common skills you should practice.

1. Finding an unknown starting value or multiplier

If you know two terms in a geometric sequence, you can solve for $r$.

Example: If $a_1=4$ and $a_3=36$, then

$$a_3=a_1r^2.$$

So,

$$36=4r^2,$$

$$9=r^2,$$

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

Both values work mathematically, but the context decides whether a negative ratio makes sense.

2. Interpreting growth in context

Suppose a city’s population increases by $2\%$ each year. The multiplier is

$$1.02.$$

If the current population is $P_0$, then after $n$ years the model is

$$P_n=P_0(1.02)^n.$$

This is geometric change and exponential growth.

3. Comparing additive and multiplicative change

If a salary increases by $1000$ dollars each year, that is arithmetic. If it increases by $3\%$ each year, that is geometric. Over short periods, the arithmetic model may look steady, but over longer periods, exponential growth can become much larger because each new increase is based on a bigger amount.

Conclusion

Arithmetic and geometric sequences describe two major kinds of change. Arithmetic sequences use a constant difference, while geometric sequences use a constant ratio. These ideas are foundational in AP Precalculus because they connect directly to exponential functions and the use of logarithms. students, if you can identify whether a pattern changes by adding or multiplying, you can choose the correct formula, model real situations, and solve many problems about growth and decay.

Study Notes

Arithmetic sequence: a sequence where consecutive terms have a constant difference $d$.
Arithmetic recursive formula: $a_n=a_{n-1}+d$.
Arithmetic explicit formula: $a_n=a_1+(n-1)d$.
Geometric sequence: a sequence where consecutive terms have a constant ratio $r$.
Geometric recursive formula: $a_n=a_{n-1}r$.
Geometric explicit formula: $a_n=a_1r^{n-1}$.
Arithmetic change is additive; geometric change is multiplicative.
A constant difference suggests an arithmetic sequence.
A constant ratio suggests a geometric sequence.
Geometric sequences connect to exponential functions because the variable appears in the exponent.
Logarithms are useful for solving equations where the unknown is in the exponent.
Real-world arithmetic examples: fixed weekly savings, constant salary raise, steady stair steps.
Real-world geometric examples: compound interest, population growth, repeated percent increase or decrease.