Recognize sequences as functions and write arithmetic and geometric sequences both recursively and explicitly (NC.M1.F-IF.3, F-BF.2).

What this topic is asking

NC.M1.F-IF.3 says sequences are functions whose domain is a subset of the integers. NC.M1.F-BF.2 asks you to write arithmetic and geometric sequences both recursively (in terms of the previous term) and explicitly (in terms of $n$), and to translate between the forms. Sequences connect to linear functions (arithmetic) and exponential functions (geometric).

Two kinds of sequence

The difference is whether you add or multiply.

To tell them apart: subtract consecutive terms (constant difference means arithmetic) or divide consecutive terms (constant ratio means geometric).

Explicit versus recursive rules

Both forms describe the same sequence but answer different questions.

• Explicit gives the $n$th term directly from $n$: best for "find the $50$th term."
• Recursive gives each term from the previous one: best for showing the step-by-step pattern.

Since NC Math 1 provides no reference sheet, memorize both explicit forms.

Writing an arithmetic sequence

Writing a geometric sequence

For $2, 6, 18, 54, \ldots$: $a_1 = 2$ and $r = 3$. The recursive rule is $a_n = a_{n-1}\cdot 3$ with $a_1 = 2$; the explicit rule is $a_n = 2\cdot 3^{\,n-1}$. The $5$th term is $a_5 = 2\cdot 3^4 = 2\cdot 81 = 162$.

How the NC Math 1 EOC examines this topic

• Gridded response. Find a specific term using the explicit rule.
• Multiple choice. Identify the common difference or ratio, or choose the correct rule.
• Technology-enhanced. Match sequences to rules, or fill in missing terms.

Sequences bridge two families: arithmetic sequences behave like linear functions (constant difference is like constant slope), while geometric sequences behave like exponential functions (constant ratio is like a growth factor). This link is central to comparing function families.

Why sequences are functions in disguise

Listing terms can hide the structure, but seeing a sequence as a function of its position reveals it. The arithmetic sequence $7, 10, 13, \ldots$ is just the linear function $a_n = 3n + 4$ evaluated at $n = 1, 2, 3, \ldots$; the geometric sequence $2, 6, 18, \ldots$ is the exponential $a_n = \frac{2}{3}\cdot 3^{n}$ at integer inputs. This is why F-IF.3 places sequences inside the Functions strand: the explicit rule is a function rule with a restricted domain. Recognizing the connection means you can use everything you know about lines and exponentials, slope as common difference, growth factor as common ratio, to analyze sequences, instead of treating them as a separate topic.

Try this

Q1. Find the $8$th term of the arithmetic sequence $4, 9, 14, \ldots$. [1 point]

• Cue. $a_1 = 4$, $d = 5$; $a_8 = 4 + 5(7) = 39$.

Q2. Write the explicit rule for the geometric sequence $5, 10, 20, \ldots$. [2 points]

• Cue. $a_1 = 5$, $r = 2$; $a_n = 5\cdot 2^{\,n-1}$.