The 42nd Period | STEM Education Hub

Use this as a quick reference for sequences as functions, common difference and common ratio, general terms, and how arithmetic and geometric growth compare.

Change in Arithmetic and Geometric Sequences infographic

🧭 Plot Summary

Unit 2 is about quantities that change by the same factor instead of the same amount. Before we meet exponential functions, this lesson starts with the cleanest version of both ideas: lists of numbers indexed by the whole numbers. A sequence is a function from the whole numbers to the real numbers, so its graph is a set of separate points. In an arithmetic sequence, every term is the one before it plus the same common difference. In a geometric sequence, every term is the one before it times the same common ratio. One adds, the other multiplies, and that single difference decides how fast the terms grow.

What you'll do in this lesson

Define a sequence as a function from the whole numbers to the real numbers, and explain why its graph is a set of discrete points.
Recognize an arithmetic sequence by its common difference, which is a constant rate of change.
Write the general term a_n = a₀ + dn, or a_n = a_k + d(n − k) when you know a term other than the first.
Recognize a geometric sequence by its common ratio, which is constant proportional change.
Write the general term g_n = g₀rⁿ, or g_n = g_kr^{n − k} when you know a term other than the first.
Compare the two: an increasing arithmetic sequence adds the same amount each step, while an increasing geometric sequence of positive values adds more each step.

Why it matters

In Topic 2.2 you will see that linear functions are arithmetic sequences with the gaps filled in, and exponential functions are geometric sequences with the gaps filled in. Savings that earn interest, populations that grow, and a car that loses value each year are all described by the ideas in this lesson. Get comfortable here and the rest of the unit builds on it.

✅ Self-Check Before You Move On

Check off each item as you get there. These aren't grades — they're your own signal.

I can explain that a sequence is a function from the whole numbers to the real numbers, and why its graph is a set of discrete points instead of a curve.I can decide whether a sequence is arithmetic by checking for a common difference (a constant rate of change) between successive terms.I can write the general term of an arithmetic sequence as a_n = a₀ + dn, or as a_n = a_k + d(n − k) from any known term.I can decide whether a sequence is geometric by checking for a common ratio (constant proportional change) between successive terms.I can write the general term of a geometric sequence as g_n = g₀rⁿ, or as g_n = g_kr^{n − k} from any known term.I can explain why an increasing arithmetic sequence adds the same amount at each step, while an increasing geometric sequence of positive values adds a larger amount at each step.

🔬 Deep Dive

Your textbook for this topic: sequences as functions, common differences and ratios, general terms, and how arithmetic and geometric growth compare — with interactive tools.

Open Deep Dive →

🧭 Guided Practice

Step through worked examples for every essential knowledge statement in 2.1.A and 2.1.B. Three examples each, revealed one step at a time.

Start Guided Practice →

📝 Assignment

20 questions covering the whole lesson. Every answer is shown, with a pull-down that walks through how it's worked.

Open Assignment →

📌 AP Exam Note

Expect to move between a list of terms, a table, a graph, a verbal context, and the rule. Justify "arithmetic" by showing the same difference between every pair of successive terms, and "geometric" by showing the same ratio. The AP framework starts counting at n = 0, so a₀ and g₀ are the initial values, and a context will often hand you a term other than the initial one.