Construct and interpret arithmetic sequences with explicit and recursive rules, and connect them to linear functions whose domain is the integers (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard covers arithmetic sequences and frames them as a special kind of linear function. An arithmetic sequence adds a fixed amount, the common difference, to get from one term to the next. The Georgia Milestones EOC asks you to write the explicit rule (a formula for the $n$th term), the recursive rule (each term from the previous one), and to find a specified term. Because the formulas are usually not on the reference sheet, this is a memorization-and-application topic, and the connection to linear functions is the conceptual payoff.

The common difference

In an arithmetic sequence, consecutive terms differ by a constant, the common difference $d$. Find it by subtracting any term from the next: for $5, 8, 11, 14, \dots$, $d = 8 - 5 = 3$. A negative common difference gives a decreasing sequence, like $20, 17, 14, \dots$ with $d = -3$.

The explicit rule

The explicit (closed) rule gives the $n$th term directly, without listing the terms before it.

The factor is $(n - 1)$, not $n$, because by the time you reach the first term you have added the difference zero times. For $5, 8, 11, \dots$: $a_n = 5 + 3(n - 1)$, so $a_{10} = 5 + 3(9) = 32$.

The recursive rule

The recursive rule defines each term from the one before it, and must include the starting value.

For the same sequence: $a_n = a_{n-1} + 3$ with $a_1 = 5$. The recursive rule is natural for "what is the next term," while the explicit rule is faster for "what is the 50th term."

An arithmetic sequence is a linear function

The explicit rule $a_n = a_1 + (n - 1)d$ rearranges to $a_n = dn + (a_1 - d)$, which is linear in $n$ with slope $d$. So an arithmetic sequence is a linear function whose domain is the positive integers rather than all real numbers. The common difference is the slope, and the terms are equally spaced points on a line.

How the Milestones examines this topic

• Multiple choice. Choose the explicit rule, with the $(n - 1)$-versus-$n$ trap among the distractors.
• Numeric entry. Find a specified term, or the common difference.
• Constructed response. Write both the explicit and recursive rules and find a term, stating the starting value.

Why the explicit rule uses (n - 1)

The single most common error in this topic is writing $a_n = a_1 + nd$ instead of $a_n = a_1 + (n - 1)d$, so it is worth understanding rather than memorizing. The first term $a_1$ is the starting point, before any common difference has been added. The second term has added $d$ once, the third has added it twice, and in general the $n$th term has added $d$ exactly $(n - 1)$ times, one fewer than its position. That is why the multiplier is $(n - 1)$. A quick check at $n = 1$ confirms it: $a_1 = a_1 + (1 - 1)d = a_1$, as it must, whereas the wrong formula would give $a_1 + d$, already one step too far. Building in this $n = 1$ check catches the error every time.

Modeling with arithmetic sequences

Arithmetic sequences model situations with a constant per-step change: a savings plan adding a fixed amount each week, seats increasing by a fixed number per theater row, a stack growing by a constant amount. Writing the model means identifying the first term (the starting amount) and the common difference (the per-step change), then using the explicit rule to answer "how much after $n$ steps." This is the same starting-value-and-rate structure as a linear model, which is why arithmetic sequences and linear functions are taught together, and recognizing the shared structure lets you reuse the linear-modeling habits on sequence problems.

Try this

Q1. Write the explicit rule for $3, 9, 15, 21, \dots$ and find $a_8$. [2 points]

• Cue. $d = 6$, $a_n = 3 + 6(n - 1)$; $a_8 = 3 + 6(7) = 45$.

Q2. A sequence has $a_1 = 100$ and $d = -5$. Write the recursive rule. [1 point]

• Cue. $a_n = a_{n-1} - 5$ with $a_1 = 100$.