Arithmetic Sequences

Introduction

An arithmetic sequence is a sequence in which the difference between every pair of consecutive terms is constant. This constant is called the common difference, usually written as $d$ . Arithmetic sequences are important in algebra because they connect patterns, linear functions, equations, graphs, and real-world situations in which a quantity changes by the same amount each step.

For example, the sequence $5, 8, 11, 14, 17, \dots$ is arithmetic because each term is 3 more than the previous term. Its common difference is $d = 3$ . The sequence $20, 16, 12, 8, \dots$ is also arithmetic, but its common difference is $d = - 4$ . A constant sequence such as $7, 7, 7, 7, \dots$ is arithmetic too, with $d = 0$ .

The image above shows a simple arithmetic sequence plotted against its term number. Notice how the points line up: equal changes in term number produce equal changes in value.

By the end of this aiMOOC, you should be able to recognize arithmetic sequences, find their common difference, write explicit and recursive rules, determine unknown terms, connect arithmetic sequences with linear graphs, solve contextual problems, and calculate sums of finite arithmetic sequences.

Core Ideas

Sequence Notation and the Common Difference

A sequence is an ordered list of numbers. The notation $a_{n}$ means the term in position $n$ . Thus, $a_{1}$ is the first term, $a_{2}$ is the second term, and so on.

For an arithmetic sequence, the common difference can be found by subtracting any term from the next term:

$d = a_{n} - a_{n - 1}$

If this difference is the same for all consecutive pairs, the sequence is arithmetic.

Example: For $14, 9, 4, - 1, - 6, \dots$ ,

$9 - 14 = - 5$ , $4 - 9 = - 5$ , and $- 1 - 4 = - 5$ .

Therefore, $d = - 5$ .

A useful habit is to check more than one pair of consecutive terms. A single matching difference is not enough to prove that an entire sequence is arithmetic.

The Explicit Formula

The first term is $a_{1}$ . To reach the second term, you add $d$ once. To reach the third term, you add $d$ twice. To reach the $n$ th term, you add $d$ exactly $n - 1$ times. This gives the explicit formula:

$a_{n} = a_{1} + (n - 1) d$

An explicit rule lets you find any term directly without calculating all earlier terms.

Example: Suppose $a_{1} = 7$ and $d = 4$ . Then

$a_{n} = 7 + (n - 1) 4 = 4 n + 3$ .

The 25th term is

$a_{25} = 7 + 24 \cdot 4 = 103$ .

You can also use the formula to decide whether a number occurs in a sequence. For $7, 11, 15, 19, \dots$ , solve $7 + (n - 1) 4 = 99$ . This gives $n = 24$ , so 99 is the 24th term.

The Recursive Formula

A recursive rule defines a term using one or more earlier terms. For an arithmetic sequence, you give the first term and then repeatedly add the common difference:

$a_{1} = A$

$a_{n} = a_{n - 1} + d$ for $n \geq 2$ .

For the sequence $7, 11, 15, 19, \dots$ , a recursive description is

$a_{1} = 7$

$a_{n} = a_{n - 1} + 4$ for $n \geq 2$ .

The explicit form is usually faster when you need a distant term. The recursive form emphasizes how the sequence grows from one term to the next.

Arithmetic Sequences as Linear Functions

If you plot the ordered pairs $(n, a_{n})$ , the points of an arithmetic sequence lie on a straight line. Starting from

$a_{n} = a_{1} + (n - 1) d$ ,

you can rewrite the rule as

$a_{n} = d n + (a_{1} - d)$ .

This has the same algebraic structure as the linear equation $y = m x + b$ . The common difference $d$ plays the role of the slope. However, an arithmetic sequence is discrete: its input values are normally positive integers $1, 2, 3, \dots$ , not every real number.

This connection helps you move between tables, formulas, and graphs. If the sequence increases, then $d > 0$ ; if it decreases, then $d < 0$ ; if it is constant, then $d = 0$ .

Solving Problems with Arithmetic Sequences

Finding Missing Terms

You can use equal differences to fill gaps.

Suppose $4, ◻, ◻, 22$ is arithmetic. There are three equal jumps from 4 to 22, so

$d = \frac{22 - 4}{3} = 6$ .

The sequence is therefore $4, 10, 16, 22$ .

More generally, if you insert $k$ arithmetic means between two numbers $x$ and $y$ , there are $k + 1$ equal jumps, so

$d = \frac{y - x}{k + 1}$ .

Finding the Position of a Term

If $d \neq 0$ , you can solve the explicit formula for $n$ :

$n = 1 + \frac{a_{n} - a_{1}}{d}$ .

The result must be a positive integer for the proposed value to be a term of the sequence.

Example: Is 86 in the sequence $2, 9, 16, 23, \dots$ ?

Here $a_{1} = 2$ and $d = 7$ . Then

$n = 1 + \frac{86 - 2}{7} = 13$ .

So 86 is the 13th term.

If $d = 0$ , every term equals $a_{1}$ , so you handle that case separately.

Real-World Modeling

Arithmetic sequences model situations with a constant additive change.

Theater seating: A theater has 18 seats in the first row, and each following row has 2 more seats. The number of seats in row $n$ is

$a_{n} = 18 + 2 (n - 1)$ .

The 20th row has $18 + 2 \cdot 19 = 56$ seats.

Weekly savings plan: You save $12 in week 1 and increase the amount by $3 each week. The weekly deposits form the arithmetic sequence $12, 15, 18, \dots$ . In week 10, you save $39$ dollars.

The key modeling question is: Does the quantity change by the same amount each step? If the change is multiplicative instead, a geometric sequence may be a better model.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. If

$S_{n} = a_{1} + a_{2} + \dots + a_{n}$ ,

then the sum of the first $n$ terms is

$S_{n} = \frac{n}{2} (a_{1} + a_{n})$ .

Using $a_{n} = a_{1} + (n - 1) d$ , you can also write

$S_{n} = \frac{n}{2} (2 a_{1} + (n - 1) d)$ .

The visual above represents a standard pairing idea. If you place an arithmetic sequence next to a reversed copy, each pair has the same total $a_{1} + a_{n}$ . There are $n$ such pairs in the doubled sum, so dividing by 2 gives the formula.

Example: The first 20 rows of a theater contain $18, 20, 22, \dots, 56$ seats. The total number of seats is

$S_{20} = \frac{20}{2} (18 + 56) = 740$ .

For the savings example $12, 15, 18, \dots, 39$ , the total saved in 10 weeks is

$S_{10} = \frac{10}{2} (12 + 39) = 255$ dollars.

Triangular Numbers as an Arithmetic Sum

The triangular numbers are formed by the sums

$1$ ,

$1 + 2$ ,

$1 + 2 + 3$ ,

and so on.

Because $1, 2, 3, \dots, n$ is an arithmetic sequence with $a_{1} = 1$ , $a_{n} = n$ , and $d = 1$ ,

$1 + 2 + \dots + n = \frac{n (n + 1)}{2}$ .

This is a useful example of how a visual pattern, a sequence, and an algebraic formula can represent the same mathematical structure.

Common Errors and How to Avoid Them

Forgetting the $n - 1$ : In $a_{n} = a_{1} + (n - 1) d$ , the first term requires zero jumps, not one.

Confusing a term with a sum: $a_{n}$ is one term; $S_{n}$ is the sum of several terms.

Using a ratio instead of a difference: Arithmetic sequences have a constant additive difference. Geometric sequences have a constant multiplicative ratio.

Treating the graph as continuous: The points of an arithmetic sequence lie on a line, but the sequence itself is normally defined only at integer term numbers.

Ignoring signs: For decreasing sequences, the common difference is negative.

Interactive Tasks

Quiz: Test Your Knowledge

Which sequence is arithmetic? (4, 9, 14, 19) (!3, 6, 12, 24) (!2, 5, 9, 14) (!1, 4, 9, 16)

What is the common difference of 17, 12, 7, 2? (-5) (!5) (!-10) (!12)

What is the explicit formula for an arithmetic sequence with first term 6 and common difference 4? (a_n = 4n + 2) (!a_n = 6n + 4) (!a_n = 6n - 2) (!a_n = 24n)

What is the 12th term of 3, 8, 13, 18, and so on? (58) (!53) (!60) (!63)

Which statement describes a recursive arithmetic rule? (Each term equals the previous term plus a constant) (!Each term equals the previous term times a constant) (!Each term equals its position squared) (!Each term equals the sum of all earlier terms)

What does the common difference represent on a graph of term number versus term value? (The slope between consecutive plotted points) (!The horizontal intercept) (!The number of plotted points) (!The product of the coordinates)

Which statement gives the sum of the first n terms of an arithmetic sequence? (Half of n times the sum of the first and last terms) (!The first term plus n times the common difference) (!The first term times the common difference to the nth power) (!The number of terms divided by the last term)

If 5, x, 17 is an arithmetic sequence, what is x? (11) (!10) (!12) (!6)

A sequence begins at 40 and decreases by 3 each term. What is its common difference? (-3) (!3) (!37) (!40)

A theater has 10 seats in the first row and 2 more seats in each new row. How many seats are in the 8th row? (24) (!22) (!26) (!80)

Memory Game

Common difference	Constant change between consecutive terms
Explicit rule	Formula that gives a term directly from its position
Recursive rule	Rule that generates a term from the previous term
Arithmetic series	Sum of terms from an arithmetic sequence
Discrete graph	Set of separate points at integer input values
First term	Value represented by a subscript of one

Drag and Drop

Match the correct terms.	Topic
Constant difference	Arithmetic sequence
Direct term formula	Explicit representation
Previous-term rule	Recursive representation
Straight-line pattern	Graph of term values
Finite term total	Arithmetic series

...

Crossword Puzzle

Difference	What constant do you find by subtracting consecutive terms?
Explicit	What type of rule finds any term directly from its position?
Recursive	What type of rule uses the previous term to generate the next?
Linear	What type of function has the same constant rate-of-change structure?
Sequence	What is an ordered list of mathematical terms called?
Arithmetic	What kind of sequence has a constant additive change?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Sequence Hunt: Find three arithmetic sequences in textbooks, timetables, games, sports scoring, or everyday routines; identify the first term and common difference of each.
Pattern Poster: Create a one-page poster that shows an arithmetic sequence as a list, table, graph, explicit rule, and recursive rule.
Human Number Line: Work with classmates to stand in positions representing consecutive terms of an arithmetic sequence and explain how the equal step size models the common difference.
Mini Tutorial Video: Record a one-minute video explaining how to test whether a sequence is arithmetic using at least two consecutive differences.

Standard

Theater Design: Design a seating plan in which each row has a fixed number of seats more than the previous row; calculate a chosen row and the total capacity.
Savings Model: Create a ten-week savings plan with arithmetic weekly deposits, calculate the final weekly deposit and total amount, and explain whether the plan is realistic.
Graph Investigation: Plot at least three arithmetic sequences with different common differences and describe how the sign and size of the common difference affect the graph.
Interview About Patterns: Interview a worker, coach, craftsperson, or family member about a situation involving regular increases or decreases; decide whether an arithmetic sequence is an appropriate model and justify your decision.

Advanced

Formula Derivation: Produce a written or visual derivation of the explicit formula from repeated addition and explain why the factor n minus one is necessary.
Series Proof Project: Create an image, animation, or physical model that demonstrates why the arithmetic-series formula works by pairing first and last terms.
Model Comparison: Investigate one arithmetic-growth situation and one geometric-growth situation, build formulas and graphs for both, and explain when the two models behave differently.
Data Modeling Challenge: Collect or generate data that are approximately linear, fit an arithmetic-sequence model to selected discrete points, analyze deviations from constant difference, and discuss the model's limitations.

Learning Assessment

Representations Assessment: Given a table from an unknown arithmetic sequence, determine its rule, graph it, and explain how the common difference appears in both representations.
Error Analysis Assessment: Analyze a solution in which a learner used a_n = a_1 + nd, identify the error, correct the formula, and demonstrate the correction with a numerical example.
Modeling Assessment: Build and solve an arithmetic-sequence model for a real-world scenario with constant additive change, stating assumptions and interpreting the result in context.
Series Reasoning Assessment: Derive or justify the arithmetic-series formula using pairing, a diagram, or algebra, then apply it to a new finite sum.
Transfer Assessment: Compare an arithmetic and a geometric sequence that share the same first two terms, and explain how their later behavior reveals the difference between additive and multiplicative change.
Decision Assessment: Given several data sets, decide which can be modeled exactly by arithmetic sequences, which are only approximate, and which require a different model; defend every choice.

Evidence of Learning

Strong evidence of learning includes accurate use of the terms sequence, term, common difference, explicit rule, recursive rule, and arithmetic series; correct calculation of distant terms and finite sums; successful movement between lists, tables, formulas, graphs, and contexts; explanations that connect the common difference with slope; and justified decisions about when an arithmetic model is or is not appropriate.

Your products can include annotated graphs, worked solutions, mathematical posters, short explanatory videos, spreadsheet models, interview notes, physical pattern models, or written proofs. High-quality work should show not only correct answers but also reasoning, clear notation, interpretation of results, checking methods, and transfer to unfamiliar problems.

OERs on the Topic

For further study, the English Wikipedia article on arithmetic progression provides definitions, formulas, examples, and extensions beyond this course.

Linked Learning Areas

Arithmetic sequences connect strongly with algebra, functions, coordinate graphs, equations, mathematical modeling, finite sums, and geometric sequences. The navigation below summarizes the most useful linked learning areas.

Arithmetic Sequences

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