Arithmetic Sequence

Sommaire

In short: An arithmetic sequence is a numbered list that climbs or falls by one constant amount per step, known as the common difference, from each term to the next.

Algebra 1Functions and GraphsF-IF.A.3

What is Arithmetic Sequence?

Count seats in an auditorium where each row holds 3 more than the row ahead of it and you get an arithmetic sequence: 14, 17, 20, 23, and onward. Neighboring entries are always separated by the same gap.

That gap is the common difference $d$, found by subtracting any entry from the one after it. It may be positive, negative or zero. The $n$th entry is $a_n = a_1 + (n – 1)d$, where $a_1$ is the first.

How to spot it

Numbers in a list rise or drop by an identical amount at every step.
A story keeps adding the same quantity, like $35 saved each month or three extra seats per row.
Plotted as dots, the terms line up along one straight line.
You are asked for the 40th entry, or for the position of a given number.

The method

Subtract consecutive terms to get the common difference d, and confirm it is the same everywhere.
Identify the first term $a_1$.
Substitute into $a_n = a_1 + (n – 1)d$ with the n you need.
Simplify, and for a reverse question set $a_n$ equal to the target and solve for n.

Each bar is 4 taller than the last: common difference d = 4.

Arithmetic Sequence: solved example

Problem

A theater’s front row holds 14 seats, and every row behind it holds 3 more than the row in front. How many seats does row 20 hold?

Solution

Here $a_1 = 14$, $d = 3$ and $n = 20$. Use $a_n = a_1 + (n – 1)d$.

$a_{20} = 14 + (20 – 1)(3) = 14 + 57 = 71$. Row 20 has 71 seats.

Try it

Question 1

Find the 15th term of $-5, -1, 3, 7, \ldots$.

Show the answer

$d = 4$, so $a_{15} = -5 + 14(4) = 51$.

Question 2

Is $3, 6, 12, 24, \ldots$ an arithmetic sequence?

Show the answer

No. The differences are 3, 6 and 12, which are not constant.

Arithmetic Sequence vs. Geometric Sequence

Here the step is a sum, so plotted terms follow a straight line. In the geometric case the step is a product, so terms swell or shrink faster and faster. Test by subtracting neighbors first, then by dividing them.

Geometric Sequence

Keep going

Lessons

Arithmetic and Geometric Sequences

Worksheets and quizzes

Related words

Whole dictionary: A–Z · Algebra 1 · Functions and Graphs

Voter.. page