Introduction

Welcome to unit 2! In this topic, you’ll learn how to represent, analyze, and compare two fundamental types of sequences: arithmetic and geometric. These sequences appear frequently in both mathematical problems and real-world contexts, such as financial planning, population modeling, and patterns of growth and decay.

What is a sequence? 

A sequence is an ordered list of numbers, whose domain is all positive integers and whose values are real numbers. The notation $a_n$ denotes the nth term (or value at input $n$) of the sequence. Because the input $n$ must be a positive integer, a sequence is discrete, meaning it only exists at specific, distinct values, which also means that nothing exists between those values/points. Typically, on a graph, functions appear as one connected and continuous line; however, if it is discrete, the graph will appear as individual points with gaps between them.

For example, if $a(n)=2n+5\text{ for }n=0,1,2,\dots$ we plot ONLY the points $(0,5),(1,7),(2,9),\dots$ Thinking of a sequence as a function on whole numbers helps us use function language; we can use terms like domain or formulas just as with other functions. However, you must keep in mind the main differences, which are A) the domain is restricted to exclusively the positive integers for sequences, and  B) The sequence is discrete and thus the graph consists only of separate points rather than a continuous line.

We will provide consistent reminders about this throughout, but Collegeboard uses a style called zero-based indexing. This means that $a_0$ is considered to be the "first term", and is written as the $a_0$. It may sometimes be called the $0$th term, but it is synonymous with the STARTING TERM of the sequence. AP Precalculus uses zero-based indexing when defining both their arithmetic and geometric sequences.

Arithmetic Sequences

An arithmetic sequence is one of the two sequences you will learn in this AP Precalculus course; it is a sequence in which successive terms differ by a constant amount. In other words, there is a fixed “d” value (the common difference) such that each term is the previous term plus d:

$a_{n+1}+d,n=0,1,2,\dots$

where n is whatever term you’re on, or in other words, the term number. Collegeboard assumes $n=0$ is the first term; some other websites may use $n=1$, or $2$, or etc. as their starting term, but for this class, you will be using @$n=0$ as your first term. This style is called zero-based indexing, and it means the first term is written as $a_0$, which we usually call the 0th term. Collegeboard also uses zero-based indexing when defining their sequences.   

In some capacity, we can view Arithmetic sequences as a linear function, but one where n (or the domain) is only defined on all the positive integers.

In order to find the common difference d (given an arithmetic sequence), we can use the fact that each consecutive term is separated by that difference d. For example, the sequence:

$5,8,11,14\dots$

Is arithmetic with a common difference $d=3$, because each consecutive term increases by a difference of $3$. Listing the terms shows the pattern clearly.

$a_0=5,a_1=5+3=8,a_2=5+2\cdot 3=11,a_3=5+3\cdot 3=14,\dots$

In general, writing out a few terms reveals that an arithmetic sequence actually has the form:

{$a_0,a_0+d,a_0+2d,a_0+3d,\dots$}

Following this pattern, after n terms (where $a_0$ is the $0$th term), we have added $d$ exactly $n$ times (starting from $a_0$), so the general ($n$th) term is

$a_n=a_0+dn$

Equivalently, if instead we know the value at some index $k$ (say $a_k$), and want the value at index $n$, we add a total of $n-k$ times:

$a_n=a_k+d(n-k)\text{ < < < MEMORIZE THIS ONE}$

EXAMPLE:

Suppose a savings account has an initial deposit of $100 at n = 0 (meaning a0=100) and then you deposit $20 at the start of each month. From here, we can see that the balance of the savings account is changing by a constant rate of $20 per month, which represents the common difference, since, every month, the balance is changing by said difference. If we write out the first few terms, it would look like this:

$100+0(20)=100,100+1(20)=120,100+2(20)=140,100+3(20)=160,\text{etc.}$

From here, it becomes more apparent that $d$ is equal to $20.

The monthly balance (in dollars) then forms an arithmetic sequence with $a_0=100+20n$, where $n$ is months.

So, after $5$ months, $a_5=100+20\cdot 5=200$. 

In summary, arithmetic sequences have equal additive changes each step. Their explicit formula is $a_n=a_k+d(n-k)$. There is an implicit formula, $a_n=a_{n-1}+d$ or $a_{n+1}=a_n+d$, but this version is rarely used.

Let’s check comprehension. Problems are listed in increasing difficulty (Answers are in separation section below):

Geometric Sequences

A geometric sequence is a sequence in which successive terms differ multiplicatively by a constant factor (known as the common ratio). That is, there is a fixed number r such that

$g_{n+1}=r\cdot g_n,n=0,1,2,\dots$

Each term is the previous term multiplied by $r$. Equivalently, each term divided by its previous term is constant. If $r_0$, then $\frac{g_n}{g_{n-1}}=r$ for all $n$. For example, the sequence

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

Is geometric with a common ratio $r=2$, since each term is twice the value of the previous term, which can be more clearly seen in below:

$3\cdot 2^0=3,3\cdot 2^1=6,3\cdot 2^2=12,3\cdot 2^3=24,\text{etc}.$

Here, you can see that each time the term number ($n$) increases by one, the value is doubled, or multiplied by two. This multiplicative effect is a result of our common ratio $r$, which represents whatever number is being multiplied to the previous term of a geometric sequence.

Writing out terms:

$g_0=3,g_1=3\cdot 2=6,g_2=3\cdot 2^2,g_3=3\cdot 2^3=24,\dots$

Shows the pattern. In general, starting from $g_0$ and multiplying $r$ an $n$ number of times gives

$g_n=g_0\cdot r^n$

More generally, if we know $g_k$ and want $g_n$, we multiply by $r$ a total of $n-k$ times:

$g_n=g_k\cdot r^{n-k}$

This formula follows because each step multiplies the previous term by $r$, so $g_1=g_0\cdot r$, $g_2=g_0\cdot r^2$

EXAMPLE:

Say there was a colony of bacteria whose population triples every hour. If there were $10$ bacteria at time 0, then after 1 hour there are $10\ast 3=30$, after 2 hours, $10\ast 3^2=90$, etc. The population sequence is $10,30,90,270,\dots$ Its formula is

$g_n=10\cdot 3^n$

Since each hour multiplies by 3. For instance, after 4 hours there are $g_4=10\cdot 3^4=810$ bacteria. Alternatively, if you know at hour 2 there are 90 bacteria ($g_2=90$) and $r=3$, then at hour 5: $g_5=g_2\cdot 3^{5-2}=90\cdot 3^3=2430$.

Thus geometric sequences show constant proportional change (multiplying by $r$ each step).

Let’s check comprehension. Once again, problems are listed in increasing difficulty (Answers are in separate section below):

Answer Keys

Arithmetic Sequences Comprehension Check

Explanation: Remember, $n$ represents the term number. If we are solving for the $12$th term, we are asking for the value of the sequence when $n=12$. We simply plug $n$ in.

$\frac{4n}{2n^3-3}⟹\frac{(4)(12)}{2(12{)}^2-3}=\frac{48}{2(144)-3}=\frac{48}{285}=\frac{16}{95}$

Explanation: Using the explicit formula, we can solve for the common difference $d$.

$a_n=a_k+d(n-k)⟹a_6=a_3+d(6-3)⟹85=94+3d⟹-9=3d⟹d=-3$

We can find the explicit form of the sequence with the $d$ value found.

$a_n=a_k+d(n-k)⟹a_n=a_k-3(n-k)$

From here, we have $2$ ($3$ if you wanna be extra) choices. We can use $a_3=94$

$a_n=a_3-3(n-3)⟹a_n=94-3(n-3)$

We can also use $a_6=85$

$a_n=a_k-3(n-k)=a_n=a_6-3(n-6)=a_n=85-3(n-6)$

If requested, you can also find the explicit form using $a_0$ after finding it.

$a_n=94-3(n-3)⟹a_0=94-3(0-3)=103$

Thus,

$a_n=a_0-3(n-0)⟹a_n=103-3n$

$4,\frac{3}{2},-1,-\frac{7}{2}$

Explanation: An arithmetic sequence’s common difference can be found by subtracting the $n$th term by the $n+1$th term. 

$a_{n+1}-a_n=\frac{3}{2}-4=\frac{-5}{2}$

This is our common difference. With this, we can simply plug in our needed values to create the formula.

$a_n=a_k+d(n-k)⟹a_n=a_0-\frac{5}{2}(n-0)⟹a_n=4-\frac{5n}{2}$

Geometric Sequences Comprehension Check

Explanation: We are given $a_1=5$, $r=\frac{3}{2}$, which is all we need to fill the skeletal formula in:

$g_n=g_k\cdot r^{n-k}⟹g_n=g_1\cdot r^{n-1}⟹g_n=5\cdot {\left(\frac{3}{2}\right)}^{n-1}$

To solve for the $8$th term, plug in $n=8$ into the equation.

$g_n=5\cdot {\left(\frac{3}{2}\right)}^{n-1}⟹g_8=5\cdot {\left(\frac{3}{2}\right)}^{8-1}=5\ast \frac{2187}{128}=\frac{10935}{128}$

Explanation: The geometric sequence can be represented as such with the current information:

$a_0,a_1,a_2,a_3,a_4$

$4,a_1,a_2,a_3,324$

We can set $a_0=4$ and $a_4=324$. We can now plug this into the skeletal formula and solve for $r$.

$g_n=g_k\cdot r^{n-k}⟹g_4=g_0\cdot r^4⟹324=4r^4⟹\pm \sqrt[4]{81}=\sqrt[4]{r^4}⟹\pm 3=r$

Because there are two possible answers for $r$, we will have to test if the answer works.

When $r=3$, the sequence follows as $4,12,36,108,324$. The sequence matches with the info given, and so $r=3$ can be the common ratio.

When $r=-3$, the sequence follows as $4,-12,36,-108,324$. The sequence also matches with the info given, and so $r=-3$ can be the common ratio too.

The three geometric means of the sequence can be $12,36,108$ when $r=3$, and $-12,36,-108$ when $r=-3$.

Explanation: We are given the implicit form of a geometric sequence, as one of the terms is multiplied by a constant value. Comparing both $a_{k+1}=-2a_k$ and the skeletal formula $g_n+1=r\cdot g_n$ makes it clear that $r=-2$.

Then, we need to find the formula for the $n$th term of the sequence. Plugging into the explicit formula, we get:

$g_n=g_k\cdot r^{n-k}=g_1\cdot r^{n-1}=5\cdot (-2{)}^{n-1}$

The formula for the $n$th term of the sequence is $g_n=5\cdot (-2{)}^{n-1}$

Conclusion

In arithmetic sequences, each term increases by the same fixed amount, which means the growth is steady and predictable. No matter how far along the sequence you go, the difference between consecutive terms remains constant. In contrast, geometric sequences grow by a consistent factor rather than a fixed amount. This means that as the sequence progresses, the amount by which each term increases becomes larger and larger. In other words, arithmetic growth adds the same step each time, while geometric growth multiplies by the same factor, causing the increases to compound and accelerate over time.