Recognize arithmetic and geometric sequences and find a term using the explicit formulas, connecting them to linear and exponential functions (LA A1: F-BF.A.2, F-IF.A.3).

What this topic is asking

Standards A1: F-BF.A.2 and F-IF.A.3 ask you to recognize arithmetic and geometric sequences and find a term using the explicit formulas. On LEAP 2025 these are Type I items, and the sequence formulas are on the reference sheet, so the credit is for identifying the type, finding the difference or ratio, and substituting correctly.

Arithmetic sequences

In an arithmetic sequence each term is the previous term plus a fixed number, the common difference $d$. Find $d$ by subtracting any term from the next.

Geometric sequences

In a geometric sequence each term is the previous term times a fixed number, the common ratio $r$. Find $r$ by dividing any term by the previous one.

Telling them apart

Check the consecutive terms:

• Constant difference (you subtract to a fixed number): arithmetic.
• Constant ratio (you divide to a fixed number): geometric.

For $4, 7, 10, 13$: differences are all $3$, arithmetic. For $3, 6, 12, 24$: ratios are all $2$, geometric.

How LEAP examines this topic

• Equation response. Find a specified term using the reference-sheet formula.
• Multiple choice. Identify the sequence type, the common difference or ratio, or a missing term.
• Drag and drop. Match sequences to their explicit rules.

A clarifying idea: the formulas use $n - 1$ because the first term ($n = 1$) has had no step applied: $a_1 = a_1 + 0 \cdot d$ and $a_1 = a_1 \cdot r^0$. The first step happens between term 1 and term 2.

Why sequences are functions of their position

A sequence is really a function whose input is the term number, which is the conceptual link F-IF.A.3 draws and the reason sequences sit in the functions module. Writing $a_n$ is the same as writing $f(n)$: you feed in a position $n$ (a positive integer) and the formula returns the term at that position. Seen this way, an arithmetic sequence is a linear function in disguise, $a_n = a_1 + (n - 1)d$ has a constant rate of change $d$ per step, exactly like slope, so its terms fall on a straight line. A geometric sequence is an exponential function in disguise, $a_n = a_1 r^{\,n-1}$ multiplies by the constant factor $r$ per step, so its terms grow or decay exponentially. This is why the "constant difference versus constant ratio" test for sequences mirrors the "common difference versus common ratio" test for linear versus exponential functions, they are the same idea applied to integer inputs. Recognizing the connection lets you carry everything you know about lines and exponentials over to sequences, and it explains why the reference sheet groups the sequence formulas near the other growth formulas.

Try this

Q1. Find the 10th term of the arithmetic sequence $2, 5, 8, \ldots$. [2 points]

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

Q2. Find the 4th term of the geometric sequence $5, 10, 20, \ldots$. [2 points]

• Cue. $a_4 = 5 \cdot 2^{4-1} = 5 \cdot 8 = 40$.