Identify arithmetic and geometric sequences, find a specified term using the explicit rule, and compute simple sums (Functions).

What this topic is asking

A sequence is an ordered list of numbers following a rule. The ACT tests the two standard kinds, arithmetic (a constant difference) and geometric (a constant ratio), asking you to find a specific term or a simple sum. Recognising which kind you have, then applying its explicit formula, is the whole task.

Arithmetic sequences

An arithmetic sequence grows by a fixed amount.

Geometric sequences

A geometric sequence grows by a fixed factor.

The common ratio is $r = \frac{a_{n+1}}{a_n}$, the same between any consecutive terms, and the $n$th term is $a_n = a_1 \cdot r^{n-1}$. For $4, 12, 36, \ldots$, $a_1 = 4$ and $r = 3$, so the 5th term is $a_5 = 4 \cdot 3^{4} = 4 \cdot 81 = 324$. Geometric sequences are exponential in $n$, which links this topic to exponential functions: the term number is the exponent (minus one).

Classifying a sequence

To decide which kind you have, test consecutive terms. If the difference is constant ($9 - 5 = 13 - 9 = 4$), it is arithmetic. If the ratio is constant ($\frac{6}{3} = \frac{12}{6} = 2$), it is geometric. Some sequences are neither, such as $1, 4, 9, 16$ (the squares), where neither the difference nor the ratio is constant. Checking both a difference and a ratio on the first few terms quickly tells you the type.

Simple sums

The ACT occasionally asks for a sum. A short arithmetic sum can be added directly, or use that the sum of $n$ terms equals $n$ times the average of the first and last terms: $S_n = n \cdot \frac{a_1 + a_n}{2}$. For $1 + 2 + \cdots + 10$, this is $10 \cdot \frac{1 + 10}{2} = 55$. For a short geometric sum, adding the listed terms directly is usually fastest on the ACT. Recognising the "average of first and last, times count" shortcut handles most arithmetic-sum questions without a long addition.

Recursive versus explicit rules

A sequence can be described two ways. A recursive rule gives the next term from the previous one, such as $a_1 = 5$, $a_n = a_{n-1} + 4$ (arithmetic) or $a_n = 2 a_{n-1}$ (geometric). An explicit rule gives any term directly from $n$, such as $a_n = 5 + (n - 1)4$. Recursive rules are easy to read but slow for a far-off term, since you must build up term by term; the explicit rule jumps straight to the answer. When a question asks for the 50th term, convert a recursive description to the explicit formula first, rather than listing 50 terms.

Sequences as functions

A sequence is really a function whose inputs are the term numbers $1, 2, 3, \ldots$ An arithmetic sequence is a linear function of $n$ (constant difference, like a slope), and a geometric sequence is an exponential function of $n$ (constant ratio, like a growth factor). This link explains the formulas: $a_n = a_1 + (n-1)d$ mirrors $y = mx + b$, and $a_n = a_1 r^{n-1}$ mirrors $y = a b^{x}$. Seeing sequences this way connects the topic to the rest of the Functions area and makes the explicit rules easier to remember.

Why the $n - 1$ matters

The single most reliable point of care is the $(n - 1)$ in both explicit formulas. Because the first term is $n = 1$, it has had no additions (arithmetic) or no multiplications (geometric) applied, so the $5$th term involves $4$ steps, not $5$. Writing $a_n = a_1 + (n - 1)d$ or $a_1 \cdot r^{n-1}$ and substituting carefully prevents the off-by-one error that is the most common mistake on these questions.

Try this

Q1. Find the 8th term of the arithmetic sequence $3, 7, 11, \ldots$ [1 point]

• Cue. $a_8 = 3 + (8 - 1)(4) = 3 + 28 = 31$.

Q2. Find the 4th term of the geometric sequence $5, 10, 20, \ldots$ [1 point]

• Cue. $a_4 = 5 \cdot 2^{3} = 5 \cdot 8 = 40$.