Recognize, extend, and write rules for arithmetic and geometric sequences, using the explicit formulas, and treat sequences as functions of the term number (Ohio F-IF.3, F-BF.2, F-LE.2).

What this topic is asking

Ohio standards F-BF.2 and F-IF.3 ask you to work with sequences: recognize whether a sequence is arithmetic or geometric, extend it, find a specific term, and write a rule. A sequence is a function whose inputs are the term numbers $1, 2, 3, \ldots$. The explicit formulas are on the reference sheet, so the skill is choosing and applying the right one. Sequences sit in the Functions category.

Telling the two apart

Look at how consecutive terms are related: a constant difference or a constant ratio.

To test, subtract consecutive terms (look for constant $d$) and divide consecutive terms (look for constant $r$).

The explicit formulas

The reference sheet gives both explicit (closed-form) rules, so you do not have to list every term.

Sequences as functions

Because the input is the term number $n$, a sequence is a function with domain the whole numbers (often $1, 2, 3, \ldots$). This is why an arithmetic sequence is essentially a linear function (constant rate of change $d$) sampled at whole numbers, and a geometric sequence is essentially an exponential function (constant ratio $r$) sampled at whole numbers.

How Ohio examines this topic

• Numeric response. Find a specific term, or the common difference or ratio.
• Multiple choice and multiple-select. Classify a sequence as arithmetic, geometric, or neither; match a sequence to its rule.
• Tables and drag and drop. Extend a sequence, or order terms.

Why the exponent is n minus one

The $n - 1$ in both formulas is the most common slip, and it has a clear reason. The first term, $a_1$, is the starting value before any step is taken, it is already term number $1$. To reach the $n$th term you take $n - 1$ steps from there: add $d$ that many times (arithmetic) or multiply by $r$ that many times (geometric). So the second term uses one step ($n - 1 = 1$), the third uses two, and so on. Writing $n$ instead of $n - 1$ adds one extra step and overshoots by a difference of $d$ or a factor of $r$. Anchoring on "the first term costs no steps" keeps the count right.

Why arithmetic is linear and geometric is exponential

Seeing the connection to function families makes sequences easier to reason about. An arithmetic sequence changes by the same amount each step, exactly the behavior of a linear function, whose graph is a straight line with slope $d$. A geometric sequence changes by the same factor each step, exactly the behavior of an exponential function $a_1 r^{\,n}$, whose graph curves. This is why the formulas resemble $mx + b$ and $ab^x$: $a_n = a_1 + (n-1)d$ is linear in $n$, and $a_n = a_1 r^{\,n-1}$ is exponential in $n$. Recognizing the family also tells you which grows faster in the long run, geometric (exponential) eventually overtakes arithmetic (linear).

Try this

Q1. Find the $8$th term of the arithmetic sequence $2, 5, 8, \ldots$. [2 points]

• Cue. $d = 3$, $a_8 = 2 + 7(3) = 23$.

Q2. What is the common ratio of $80, 40, 20, 10, \ldots$? [1 point]

• Cue. $40/80 = \frac{1}{2}$, so $r = \frac{1}{2}$.