2.1 Change in Arithmetic and Geometric Sequences

A sequence is a function that takes whole numbers and outputs real numbers, so its graph is a set of separate points, not a smooth curve. Arithmetic sequences grow by adding the same number each step, called the common difference $d$, while geometric sequences grow by multiplying by the same number each step, called the common ratio $r$.

Why This Matters for the AP Precalculus Exam

This topic is the foundation for all of Unit 2, which is one of the more heavily weighted parts of the AP Precalculus exam. Arithmetic sequences set up the thinking behind linear functions, and geometric sequences set up the thinking behind exponential functions. If you can recognize whether a pattern changes by a constant difference or a constant ratio, you will move faster on later topics like exponential models and logarithmic functions.

On the exam, you may need to:

• Read a table or word problem and decide if a pattern is arithmetic or geometric.
• Write the general term and use it to find a specific term.
• Recognize that sequence graphs are discrete points, not connected curves.
• Compare how arithmetic and geometric sequences grow over equal steps.

Showing clear steps is important for clear exam work, especially on free-response questions where answers without supporting work may not support a stronger score.

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Key Takeaways

• A sequence is a function from the whole numbers to the real numbers, so its graph is made of discrete points.
• Arithmetic sequences have a common difference d (constant rate of change). General term: $a_n = a_0 + dn$ or $a_n = a_k + d(n-k)$.
• Geometric sequences have a common ratio r (constant proportional change). General term: $g_n = g_0 r^n$ or $g_n = g_k r^{n-k}$.
• Add to move between arithmetic terms; multiply to move between geometric terms.
• Increasing arithmetic sequences grow by the same amount each step; increasing geometric sequences grow by a larger amount each step.
• Watch your indexing: $a_0$ is the initial term in these formulas, not $a_1$.

Sequences as Functions

A sequence is a function from the whole numbers to the real numbers. It takes a whole number (like n) and assigns it a real number. Think of a class schedule where Monday is 1, Tuesday is 2, and so on. Each position gets one output value.

Because the inputs are only whole numbers, the graph of a sequence is a set of discrete points, not a smooth curve. For example, you could track hours of sleep each night: 8 hours on day 1, 6 hours on day 2, 7 hours on day 3, and so on. The input is the day and the output is the number of hours. Graphing this gives separate points, one per day.

Do not connect the dots like you would for a continuous function. The points stand alone.

Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. That constant is the common difference, d. Because the gap between terms is always the same, an arithmetic sequence has a constant rate of change.

For example, an arithmetic sequence with a first term of 5 and a common difference of 3 looks like this: 5, 8, 11, 14, 17, 20, ... . The rate of change is 3 every step.

The sign of d tells you the direction:

• If d is positive, the terms increase.
• If d is negative, the terms decrease.

Formula and Example

The general term of an arithmetic sequence with common difference d is denoted by $a_n$ and given by:

$a_n = a_0 + dn$

Here $a_0$ is the initial value, and n is the term number, so you can plug in any whole number for n to find that term.

You can also write the general term relative to a known kth term:

$a_n = a_k + d(n - k)$

This is useful when you know some term in the middle of the sequence instead of the starting value.

For a sequence with $a_0 = 5$ and $d = 3$:

• Using the first form: $a_5 = a_0 + dn = 5 + 3(5) = 20$.
• Using the second form with the known term $a_3 = 14$: $a_5 = a_3 + d(5 - 3) = 14 + 3(2) = 20$.

Both forms give the same value. The second form just lets you start from any known term instead of the initial value.

Geometric Sequences

A geometric sequence is a sequence where the ratio of any two consecutive terms is constant. That constant is the common ratio, r, which can be a fraction or a decimal. Because the ratio between terms is always the same, a geometric sequence has a constant proportional change.

For example, a geometric sequence with a first term of 2 and a common ratio of 3 looks like this: 2, 6, 18, 54, 162, ... . Each term is 3 times the one before it.

The value of r tells you the behavior:

• If r is greater than 1, the terms increase.
• If r is between 0 and 1, the terms decrease toward zero.

Formula

The general term of a geometric sequence with common ratio r is denoted by $g_n$ and given by:

$g_n = g_0 r^n$

Here $g_0$ is the initial value, and n is the term number.

You can also write the general term relative to a known kth term:

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

This works the same way as the arithmetic version: if you know one term and the common ratio, you can find any other term.

Comparing Arithmetic and Geometric Growth

The key difference is addition versus multiplication.

• Arithmetic sequences increase by the same amount each step. With a first term of 5 and d = 3, the terms 5, 8, 11, 14, 17, 20 each go up by 3.
• Geometric sequences increase by a larger amount each step (when increasing). With a first term of 2 and r = 3, the terms 2, 6, 18, 54, 162 jump by more and more each time, because each term is multiplied by 3.

This is why increasing geometric sequences eventually pull far ahead of increasing arithmetic sequences. The arithmetic pattern climbs steadily, but the geometric pattern speeds up.

How to Use This on the AP Precalculus Exam

Identify the Type First

Before using any formula, decide whether the pattern is arithmetic or geometric:

• Subtract consecutive terms. If the difference is constant, it is arithmetic.
• Divide consecutive terms. If the ratio is constant, it is geometric.

If neither the differences nor the ratios are constant, it is some other kind of sequence.

Choose the Right Formula Form

• Use $a_n = a_0 + dn$ or $g_n = g_0 r^n$ when you know the initial value.
• Use $a_n = a_k + d(n-k)$ or $g_n = g_k r^{n-k}$ when you only know a term in the middle and the difference or ratio.

Problem Solving

When a question gives you two terms, find d or r first, then back out the initial value if you need it. For an arithmetic sequence, set up the difference between term positions. For a geometric sequence, divide the terms to isolate r.

Write out each step. On free-response questions, answers without supporting work may not be accepted, and these formulas reward careful, step-by-step setup.

Common Trap

Pay attention to whether the problem indexes from $a_0$ or $a_1$. The formulas here use $a_0$ as the initial value, so an off-by-one mistake will shift every term.

Common Misconceptions

• Connecting the points: A sequence graph is discrete points only. Do not draw a continuous curve through them.
• Mixing up d and r: A common difference means you add the same number each step (arithmetic). A common ratio means you multiply by the same number each step (geometric).
• Assuming r greater than 1 always: A geometric sequence can have a ratio between 0 and 1, which makes it decrease toward zero, or even a negative ratio, which makes the signs alternate.
• Confusing the index: $a_0$ in these formulas is the initial term, not the first counted term $a_1$. Check whether n starts at 0 or 1 before plugging in.
• Thinking geometric always beats arithmetic immediately: An increasing geometric sequence pulls ahead over time, but it can start out smaller than an arithmetic sequence depending on the initial values.

Related AP Precalculus Guides

• 2.9 Logarithmic Expressions
• 2.11 Logarithmic Functions
• 2.5 Exponential Function Context and Data Modeling
• 2.8 Inverse Functions
• 2.12 Logarithmic Function Manipulation
• 2.4 Exponential Function Manipulation