How do you tell an arithmetic sequence from a geometric sequence?

Check consecutive terms two ways. If the difference between neighbors is constant, it is arithmetic; if the quotient (ratio) between neighbors is constant, it is geometric. Arithmetic sequences add the same amount; geometric sequences multiply by the same factor.

What is the formula for the nth term of a sequence on the ACT?

For an arithmetic sequence, a_n = a_1 + (n - 1)d, where d is the common difference. For a geometric sequence, a_n = a_1 · r^(n-1), where r is the common ratio. Both use the exponent or multiplier n - 1, not n.

How do you add up a sequence quickly?

For a finite arithmetic series use S_n = (n/2)(a_1 + a_n): average the first and last terms and multiply by the number of terms. For a finite geometric series use S_n = a_1(1 - r^n)/(1 - r).

Does the ACT test infinite series?

The ACT focuses on finite sequences and sums — finding a specific term or adding a set number of terms. Infinite geometric series occasionally appear, but the vast majority of questions ask for the nth term or a finite total.

CramClub

Sequences and Series

Master arithmetic and geometric sequences and their sums for the ACT Math section, including nth-term formulas and finite-series shortcuts.

ACTmathmedium18 min read6 worked examples6 practice

Key Formulas

Arithmetic nth term

$a_n = a_1 + (n - 1)d$

$d$ is the common difference; $n - 1$ steps from the first term.

Arithmetic series sum

$S_n = \dfrac{n}{2}\,(a_1 + a_n)$

Average of the first and last term, times the number of terms.

Geometric nth term

$a_n = a_1 \cdot r^{\,n-1}$

$r$ is the common ratio; the exponent is $n - 1$.

Geometric series sum

$S_n = a_1 \cdot \dfrac{1 - r^{\,n}}{1 - r}$

Valid when $r \ne 1$; works for any finite geometric sum.

Video Lesson

Explanation

Patterns With a Rule

A sequence is an ordered list of numbers, each called a term. On the ACT Math section, nearly every sequence question falls into one of two families: arithmetic (you add the same amount each step) or geometric (you multiply by the same factor each step). The whole topic collapses into a handful of formulas once you can tell the two apart, so the first move on any sequence problem is to ask: am I adding or multiplying to get the next term?

Consider $3, 7, 11, 15, \dots$ — each term is $4$ more than the last, so this is arithmetic with common difference $d = 4$ . Now look at $3, 6, 12, 24, \dots$ — each term is $2$ times the last, so this is geometric with common ratio $r = 2$ . Same first term, completely different behavior.

Arithmetic Sequences

In an arithmetic sequence the term-to-term jump is constant. If the first term is $a_1$ and the common difference is $d$ , the $n$ th term is:

$a_n = a_1 + (n - 1)d$

The $(n-1)$ matters: to reach the 10th term you take only nine steps of size $d$ away from the first term. For $3, 7, 11, \dots$ the 20th term is $a_{20} = 3 + (20 - 1)(4) = 3 + 76 = 79$ .

To add up the first $n$ terms of an arithmetic sequence (an arithmetic series), average the first and last terms and multiply by how many terms there are:

$S_n = \frac{n}{2}\,(a_1 + a_n)$

This works because the terms are evenly spaced, so their average is exactly the midpoint of the first and last.

Geometric Sequences

In a geometric sequence you multiply by a fixed ratio $r$ each step. The $n$ th term is:

$a_n = a_1 \cdot r^{\,n-1}$

Again the exponent is $n - 1$ , not $n$ . For $5, 10, 20, 40, \dots$ the 6th term is $a_6 = 5 \cdot 2^{5} = 5 \cdot 32 = 160$ .

The sum of the first $n$ terms of a geometric series is:

$S_n = a_1 \cdot \frac{1 - r^{\,n}}{1 - r}, \quad r \ne 1$

For $2 + 6 + 18 + 54$ (four terms, $a_1 = 2$ , $r = 3$ ): $S_4 = 2 \cdot \dfrac{1 - 3^4}{1 - 3} = 2 \cdot \dfrac{1 - 81}{-2} = 2 \cdot 40 = 80$ .

Telling the Two Apart

Feature	Arithmetic	Geometric
Step operation	Add $d$	Multiply by $r$
$n$ th term	$a_1 + (n-1)d$	$a_1 \cdot r^{\,n-1}$
Graph shape	Straight line	Curved (exponential)
Test the ratio	Differences equal	Quotients equal

To decide which family a sequence belongs to, check both the differences and the quotients of consecutive terms. If the differences match, it is arithmetic; if the quotients match, it is geometric.

Working Backward for a Hidden Term

The ACT often gives you two non-adjacent terms and asks for the first term or the common difference. Treat each fact as an equation. If the 4th term is $17$ and the 9th term is $37$ in an arithmetic sequence, then from $a_9 - a_4 = (9 - 4)d$ you get $37 - 17 = 5d$ , so $d = 4$ ; then $a_4 = a_1 + 3d$ gives $17 = a_1 + 12$ , so $a_1 = 5$ .

A Reliable Game Plan

Identify the family first. Equal differences $\Rightarrow$ arithmetic; equal ratios $\Rightarrow$ geometric.
Use $n - 1$ , never $n$ , in both nth-term formulas.
Two terms given? Subtract positions to find $d$ (or divide to find $r$ ), then back-solve for $a_1$ .
A finite sum? Arithmetic uses average $\times$ count; geometric uses the ratio formula.
Count terms carefully — from the 5th to the 12th term is $12 - 5 = 7$ steps but $8$ terms.

Sequences reward calm bookkeeping more than clever insight: name $a_1$ , $d$ or $r$ , and $n$ , plug into the right formula, and the answer follows.

Worked Examples

— follow the steps, hardest last

Find a term in an arithmetic sequence

easy

The first term of an arithmetic sequence is $6$ and the common difference is $5$ . What is the 12th term?

Find a term in a geometric sequence

easy

A geometric sequence begins $4, 12, 36, \dots$ What is the 6th term?

Sum a finite arithmetic series

medium

Find the sum of the first $30$ positive even integers: $2 + 4 + 6 + \dots + 60$ .

Recover the first term from two given terms

medium

In an arithmetic sequence the 5th term is $23$ and the 11th term is $53$ . Find the first term.

Sum a finite geometric series

hard

Find the sum $3 + 6 + 12 + 24 + 48 + 96$ .

Word problem — accumulating growth

hard

A gym starts with $80$ members in its first month and gains members so that each month it has $1.5$ times as many members as the month before. To the nearest whole member, how many members does it have in its 5th month?

Watch Out For

Using $n$ instead of $n - 1$ as the exponent or multiplier, so the 10th term is off by one step.
Both nth-term formulas use $n - 1$ : reaching the 10th term takes only 9 steps away from the first term.
Confusing arithmetic and geometric — adding when you should multiply, or vice versa.
Test both: if consecutive differences are equal it is arithmetic; if consecutive quotients are equal it is geometric.
Miscounting the number of terms in a series, e.g. treating "5th through 12th term" as $12 - 5 = 7$ terms.
The count of terms from position $p$ to position $q$ inclusive is $q - p + 1$ ; the number of steps between them is $q - p$ .
Adding every term by hand on a long series and running out of time.
Use the closed-form sum formulas: average $\times$ count for arithmetic, the ratio formula for geometric.

Practice Problems

The arithmetic sequence $8, 13, 18, 23, \dots$ continues with the same pattern. What is the 10th term?

In the geometric sequence $2, 6, 18, 54, \dots$ , what is the 5th term?

What is the sum of the first $25$ terms of the arithmetic sequence whose first term is $4$ and whose common difference is $3$ ?

In an arithmetic sequence, the 3rd term is $10$ and the 7th term is $26$ . What is the first term?

What is the value of the finite geometric series $5 + 15 + 45 + 135 + 405$ ?

A ball is dropped and each bounce reaches $60\%$ of the previous height. If the first bounce reaches $50$ cm, which expression gives the height, in centimeters, of the 4th bounce?

Frequently Asked Questions

How do you tell an arithmetic sequence from a geometric sequence?: Check consecutive terms two ways. If the difference between neighbors is constant, it is arithmetic; if the quotient (ratio) between neighbors is constant, it is geometric. Arithmetic sequences add the same amount; geometric sequences multiply by the same factor.
What is the formula for the nth term of a sequence on the ACT?: For an arithmetic sequence, a_n = a_1 + (n - 1)d, where d is the common difference. For a geometric sequence, a_n = a_1 · r^(n-1), where r is the common ratio. Both use the exponent or multiplier n - 1, not n.
How do you add up a sequence quickly?: For a finite arithmetic series use S_n = (n/2)(a_1 + a_n): average the first and last terms and multiply by the number of terms. For a finite geometric series use S_n = a_1(1 - r^n)/(1 - r).
Does the ACT test infinite series?: The ACT focuses on finite sequences and sums — finding a specific term or adding a set number of terms. Infinite geometric series occasionally appear, but the vast majority of questions ask for the nth term or a finite total.

Keep Going

Functions and Graph Transformations

Evaluate functions, read function notation, and shift, stretch, and reflect graphs for the ACT Math section.

Powers and Roots

Master exponent rules, radicals, rational and negative exponents, and simplifying roots for the ACT Math section.

Linear Equations and Inequalities

Solve, model, and graph linear equations and inequalities for the ACT Math section, including slope, intercepts, and word-problem setups.

Ready to test yourself?

Put this skill to work on a full-length adaptive practice test.

Sequences and Series

Master arithmetic and geometric sequences and their sums for the ACT Math section, including nth-term formulas and finite-series shortcuts.

ACTmathmedium18 min read6 worked examples6 practice

Key Formulas

Arithmetic nth term

$a_n = a_1 + (n - 1)d$

$d$ is the common difference; $n - 1$ steps from the first term.

Arithmetic series sum

$S_n = \dfrac{n}{2}\,(a_1 + a_n)$

Average of the first and last term, times the number of terms.

Geometric nth term

$a_n = a_1 \cdot r^{\,n-1}$

$r$ is the common ratio; the exponent is $n - 1$.

Geometric series sum

$S_n = a_1 \cdot \dfrac{1 - r^{\,n}}{1 - r}$

Valid when $r \ne 1$; works for any finite geometric sum.

Video Lesson

Explanation

Patterns With a Rule

Arithmetic Sequences

In an arithmetic sequence the term-to-term jump is constant. If the first term is $a_1$ and the common difference is $d$ , the $n$ th term is:

$a_n = a_1 + (n - 1)d$

The $(n-1)$ matters: to reach the 10th term you take only nine steps of size $d$ away from the first term. For $3, 7, 11, \dots$ the 20th term is $a_{20} = 3 + (20 - 1)(4) = 3 + 76 = 79$ .

To add up the first $n$ terms of an arithmetic sequence (an arithmetic series), average the first and last terms and multiply by how many terms there are:

$S_n = \frac{n}{2}\,(a_1 + a_n)$

This works because the terms are evenly spaced, so their average is exactly the midpoint of the first and last.

Geometric Sequences

In a geometric sequence you multiply by a fixed ratio $r$ each step. The $n$ th term is:

$a_n = a_1 \cdot r^{\,n-1}$

Again the exponent is $n - 1$ , not $n$ . For $5, 10, 20, 40, \dots$ the 6th term is $a_6 = 5 \cdot 2^{5} = 5 \cdot 32 = 160$ .

The sum of the first $n$ terms of a geometric series is:

$S_n = a_1 \cdot \frac{1 - r^{\,n}}{1 - r}, \quad r \ne 1$

For $2 + 6 + 18 + 54$ (four terms, $a_1 = 2$ , $r = 3$ ): $S_4 = 2 \cdot \dfrac{1 - 3^4}{1 - 3} = 2 \cdot \dfrac{1 - 81}{-2} = 2 \cdot 40 = 80$ .

Telling the Two Apart

Feature	Arithmetic	Geometric
Step operation	Add $d$	Multiply by $r$
$n$ th term	$a_1 + (n-1)d$	$a_1 \cdot r^{\,n-1}$
Graph shape	Straight line	Curved (exponential)
Test the ratio	Differences equal	Quotients equal

To decide which family a sequence belongs to, check both the differences and the quotients of consecutive terms. If the differences match, it is arithmetic; if the quotients match, it is geometric.

Working Backward for a Hidden Term

A Reliable Game Plan

Identify the family first. Equal differences $\Rightarrow$ arithmetic; equal ratios $\Rightarrow$ geometric.
Use $n - 1$ , never $n$ , in both nth-term formulas.
Two terms given? Subtract positions to find $d$ (or divide to find $r$ ), then back-solve for $a_1$ .
A finite sum? Arithmetic uses average $\times$ count; geometric uses the ratio formula.
Count terms carefully — from the 5th to the 12th term is $12 - 5 = 7$ steps but $8$ terms.

Sequences reward calm bookkeeping more than clever insight: name $a_1$ , $d$ or $r$ , and $n$ , plug into the right formula, and the answer follows.

Worked Examples

— follow the steps, hardest last

Find a term in an arithmetic sequence

easy

The first term of an arithmetic sequence is $6$ and the common difference is $5$ . What is the 12th term?

Find a term in a geometric sequence

easy

A geometric sequence begins $4, 12, 36, \dots$ What is the 6th term?

Sum a finite arithmetic series

medium

Find the sum of the first $30$ positive even integers: $2 + 4 + 6 + \dots + 60$ .

Recover the first term from two given terms

medium

In an arithmetic sequence the 5th term is $23$ and the 11th term is $53$ . Find the first term.

Sum a finite geometric series

hard

Find the sum $3 + 6 + 12 + 24 + 48 + 96$ .

Word problem — accumulating growth

hard

Watch Out For

Using $n$ instead of $n - 1$ as the exponent or multiplier, so the 10th term is off by one step.
Both nth-term formulas use $n - 1$ : reaching the 10th term takes only 9 steps away from the first term.
Confusing arithmetic and geometric — adding when you should multiply, or vice versa.
Test both: if consecutive differences are equal it is arithmetic; if consecutive quotients are equal it is geometric.
Miscounting the number of terms in a series, e.g. treating "5th through 12th term" as $12 - 5 = 7$ terms.
The count of terms from position $p$ to position $q$ inclusive is $q - p + 1$ ; the number of steps between them is $q - p$ .
Adding every term by hand on a long series and running out of time.
Use the closed-form sum formulas: average $\times$ count for arithmetic, the ratio formula for geometric.

Practice Problems

The arithmetic sequence $8, 13, 18, 23, \dots$ continues with the same pattern. What is the 10th term?

In the geometric sequence $2, 6, 18, 54, \dots$ , what is the 5th term?

What is the sum of the first $25$ terms of the arithmetic sequence whose first term is $4$ and whose common difference is $3$ ?

In an arithmetic sequence, the 3rd term is $10$ and the 7th term is $26$ . What is the first term?

What is the value of the finite geometric series $5 + 15 + 45 + 135 + 405$ ?

A ball is dropped and each bounce reaches $60\%$ of the previous height. If the first bounce reaches $50$ cm, which expression gives the height, in centimeters, of the 4th bounce?

Frequently Asked Questions

How do you tell an arithmetic sequence from a geometric sequence?: Check consecutive terms two ways. If the difference between neighbors is constant, it is arithmetic; if the quotient (ratio) between neighbors is constant, it is geometric. Arithmetic sequences add the same amount; geometric sequences multiply by the same factor.
What is the formula for the nth term of a sequence on the ACT?: For an arithmetic sequence, a_n = a_1 + (n - 1)d, where d is the common difference. For a geometric sequence, a_n = a_1 · r^(n-1), where r is the common ratio. Both use the exponent or multiplier n - 1, not n.
How do you add up a sequence quickly?: For a finite arithmetic series use S_n = (n/2)(a_1 + a_n): average the first and last terms and multiply by the number of terms. For a finite geometric series use S_n = a_1(1 - r^n)/(1 - r).
Does the ACT test infinite series?: The ACT focuses on finite sequences and sums — finding a specific term or adding a set number of terms. Infinite geometric series occasionally appear, but the vast majority of questions ask for the nth term or a finite total.

Keep Going

Functions and Graph Transformations

Evaluate functions, read function notation, and shift, stretch, and reflect graphs for the ACT Math section.

Powers and Roots

Master exponent rules, radicals, rational and negative exponents, and simplifying roots for the ACT Math section.

Linear Equations and Inequalities

Solve, model, and graph linear equations and inequalities for the ACT Math section, including slope, intercepts, and word-problem setups.

Ready to test yourself?

Put this skill to work on a full-length adaptive practice test.