In AP Precalculus, a sequence is a function whose inputs are whole numbers (0, 1, 2, ...) and whose outputs are real numbers, so its graph is a set of discrete points instead of a continuous curve. Arithmetic sequences have a common difference; geometric sequences have a common ratio (Topic 2.1).
A sequence is just a function with a restricted domain. Instead of plugging in any real number, you only plug in whole numbers: 0, 1, 2, 3, and so on. That's why the graph of a sequence is a collection of separate dots rather than a smooth curve. The nth dot is the nth term.
AP Precalculus cares about two specific types. An arithmetic sequence changes by a constant amount each step (the common difference d), with general term a_n = a_0 + dn or a_n = a_k + d(n-k). A geometric sequence changes by a constant factor each step (the common ratio r), with general term g_n = g_0 r^n or g_n = g_k r^(n-k). Read those formulas again and you'll notice something. The arithmetic formula is a linear function in disguise, and the geometric formula is an exponential function in disguise. That parallel is the whole point of Topic 2.1.
Sequences live in Topic 2.1 (Unit 2), where learning objectives AP Pre Calc 2.1.A and 2.1.B ask you to express arithmetic and geometric sequences as functions of the whole numbers. But they also sneak into Topic 1.3 (Unit 1), where AP Pre Calc 1.3.A has you find average rates of change for sequences alongside linear and quadratic functions. The big idea sequences unlock is how a function changes. Arithmetic means equal jumps every step (constant rate of change). Geometric means proportional jumps, so an increasing geometric sequence grows by a larger amount with each successive step. The exam uses sequences as the bridge between linear thinking and exponential thinking, which is the central storyline of Unit 2.
Keep studying AP® Precalculus Unit 2
Quadratic function (Unit 1)
Take a quadratic function and compute its average rates of change over consecutive equal-length intervals. Those rates form an arithmetic sequence with a common difference. That's exactly what EK 1.3.A.2 means when it says the rates 'can be given by a linear function,' and it's one of the most-tested patterns in Topic 1.3.
Rates of change in linear functions (Unit 1)
An arithmetic sequence is a linear function sampled at whole numbers. The common difference d is the slope. If you can find slope, you can find a common difference, because they're the same idea wearing different outfits.
Exponential functions (Unit 2)
A geometric sequence is an exponential function sampled at whole numbers, with the common ratio r playing the role of the base. Topic 2.1 uses geometric sequences to set up everything Unit 2 does with exponential growth and decay.
Sequences show up in multiple-choice questions two main ways. First, the Topic 2.1 version: identify whether a table or context is arithmetic or geometric, then write or use the general term formula (a_n = a_0 + dn or g_n = g_0 r^n), often with the a_k or g_k version when you're not given the initial value. Second, the Topic 1.3 crossover: questions give you a quadratic function and ask about the sequence of average rates of change over consecutive intervals. For example, if a quadratic's average rates over [0,2], [2,4], [4,6], [6,8] are -7, -3, 1, 5, you spot the common difference of 4 and predict the next rate is 9. You need to do three things fluently: classify the sequence type, extract d or r from any two terms, and write the general term as a function of n.
A sequence IS a function, but its domain is only the whole numbers, so its graph is dots, not a curve. An arithmetic sequence looks like a linear function and a geometric sequence looks like an exponential function, but you can't evaluate a sequence at n = 2.5. On the exam, the discrete domain is what makes something a sequence.
A sequence is a function from the whole numbers to the real numbers, so its graph is discrete points instead of a continuous curve.
Arithmetic sequences have a common difference d and follow a_n = a_0 + dn, which makes them linear functions with a restricted domain.
Geometric sequences have a common ratio r and follow g_n = g_0 r^n, which makes them exponential functions with a restricted domain.
The average rates of change of a quadratic function over consecutive equal-length intervals form an arithmetic sequence, a pattern tested heavily in Topic 1.3.
An increasing arithmetic sequence grows by the same amount every step, while an increasing geometric sequence grows by a larger amount with each successive step.
The alternate formulas a_n = a_k + d(n-k) and g_n = g_k r^(n-k) let you build the general term from any known term, not just the initial value.
A sequence is a function whose domain is the whole numbers (0, 1, 2, ...) and whose outputs are real numbers, so its graph is discrete points rather than a curve. AP Precalc focuses on arithmetic sequences (common difference) and geometric sequences (common ratio) in Topic 2.1.
Yes, a sequence is a function, just one with a restricted domain. The CED defines it as a function from the whole numbers to the real numbers, which is why you can write a_n = a_0 + dn as a formula in n the same way you'd write f(x) in x.
Arithmetic sequences add the same amount each step (common difference d, formula a_n = a_0 + dn), while geometric sequences multiply by the same factor each step (common ratio r, formula g_n = g_0 r^n). Arithmetic matches linear behavior; geometric matches exponential behavior.
No. Sequences are formally defined in Topic 2.1, but Topic 1.3 in Unit 1 already uses them, since LO 1.3.A asks you to find average rates of change for sequences, and a quadratic's rates of change over equal intervals form an arithmetic sequence.
Use the kth-term versions of the formulas. For arithmetic, a_n = a_k + d(n-k); for geometric, g_n = g_k r^(n-k). Any known term plus the common difference or ratio is enough to reach any other term.
Connect this key term to the AP exam workflow: review the course, practice questions, and check related study tools.
Review units, study guides, and course resources.
Check this vocabulary in multiple-choice context.
Apply key concepts in written AP responses.
Estimate the exam score you are working toward.
Review the highest-yield facts before practice.
Put the full course together before test day.