Write arithmetic and geometric sequences both recursively and explicitly, and recognize sequences as functions on the integers (TN A1.F.BF.A.2, A1.F.IF.B.3).

What this topic is asking

Standard A1.F.BF.A.2 asks you to write arithmetic and geometric sequences both explicitly (a formula for the $n$th term) and recursively (each term from the previous one). A1.F.IF.B.3 notes that sequences are functions whose domain is the integers. The Math EOC reference sheet provides the explicit formulas, so the skill is identifying $a_1$, $d$ or $r$, and substituting.

Arithmetic versus geometric

The first move is to decide which type by checking consecutive terms.

• Arithmetic: a constant difference between terms ($+d$ each step). Example $4, 7, 10, 13$ has $d = 3$.
• Geometric: a constant ratio between terms ($\times r$ each step). Example $4, 8, 16, 32$ has $r = 2$.

Explicit and recursive forms

The explicit form computes any term directly from $n$; the recursive form needs the previous term. Both require the first term $a_1$.

Sequences as functions

A1.F.IF.B.3 frames a sequence as a function with input $n$ (a positive integer) and output $a_n$. This is why an arithmetic sequence graphs as evenly spaced points on a line (slope $d$) and a geometric sequence as points on an exponential curve (base $r$). The link is exact: arithmetic is linear with the difference as slope, geometric is exponential with the ratio as base.

How TNReady examines this topic

• Numeric response. Find a specific term ($a_{20}$) or the common difference or ratio.
• Multiple choice. Choose the explicit or recursive rule, with arithmetic-versus-geometric and $n$-versus-$(n-1)$ distractors.
• Drag and drop. Match sequences to their rules.

A clarifying idea is that arithmetic sequences and linear functions share the constant-difference, constant-slope structure, while geometric sequences and exponential functions share the constant-ratio, constant-base structure. The sequence is just the function sampled at the integers.

Why the formula uses n minus 1

The $(n - 1)$ in both explicit forms is the most-missed detail, and it has a clear reason: the first term should come out when $n = 1$, with no difference or ratio applied yet. For arithmetic, $a_1 = a_1 + (1 - 1)d = a_1 + 0 = a_1$, correct; using $n$ instead would add one extra $d$ and start the sequence at the wrong value. The same logic holds for geometric: $a_1 = a_1 r^{1-1} = a_1 r^0 = a_1$, since any base to the zero power is $1$. So $(n - 1)$ counts the number of steps taken from the first term, which is one fewer than the term number. Holding this idea, "steps from the start, not the term number", prevents the off-by-one error that the distractors are designed to catch.

Try this

Q1. Find the $10$th term of $2, 5, 8, 11, \dots$. [1 point]

• Cue. $d = 3$, $a_{10} = 2 + (10 - 1)(3) = 2 + 27 = 29$.

Q2. Write the explicit rule for the geometric sequence $5, 15, 45, \dots$. [2 points]

• Cue. $r = 3$, $a_n = 5(3)^{n-1}$.