Arithmetic Sequences

The arithmetic sequence is a pattern of numbers in which the difference between any two consecutive terms stays the same. A sequence is a collection of numbers that follows a rule or a pattern. For example, the sequence 1, 6, 11, 16,.. is an arithmetic sequence because there is a pattern where each number is obtained by adding 5 to the previous term, and this pattern is assumed to continue forever.

Examples of arithmetic sequences

There are all types of arithmetic sequences. Some of them are shown below.

$5, 8, 11, 14, 17, 20 ...$
$80, 75, 70, 65, 60, 55 ...$
$\frac{1}{2} π, π, \frac{3}{2} π, 2 π ...$
$0, - \sqrt{2}, - 2 \sqrt{2}, - 3 \sqrt{2}, - 4 \sqrt{2} ...$

The common difference in arithmetic sequences

The common difference of an arithmetic sequence is the constant difference between consecutive terms. A common difference can be a negative or positive number.

Example 1

What is the common difference in the following sequence?

$10, 21, 32, 43, 54 ...$

The common difference is 11, as each consecutive term can be reached by adding 11 to the previous term.

Example 2

What is the common difference in the following sequence?

$24, 20, 16, 12, 8 ...$

The common difference is -4 because each consecutive term can be reached by adding -4 to the previous term.

Arithmetic sequence formulas

When working with arithmetic sequences, we use the term "n" to indicate the term number and " $a_{n}$ " to indicate the $n^{th}$ term.

So, if you need to find $a_{4}$ in the sequence $1, 3, 5, 7, 9 ...$ the answer is 7, because 7 is the fourth term in the sequence.

Similarly, for any term number $n$ , ( $a_{n - 1}$ ) represents the term that comes before the $n^{th}$ term.

Recursive formulas of arithmetic sequences

Recursive formulas of arithmetic sequences provide two pieces of information:

The first term of a sequence
The pattern rule to find any term in the sequence from the term that comes before it.

For example, $a_{1} = 3$

This tells us that the first term is 3

$a_{n} = a_{n - 1} + 2$

This tells us that the $n^{th}$ term is found by adding 2 to the previous term.

Example 3

So to find the 5th term, for example, we need to extend the sequence term by term:

$a_{n} = a_{n - 1} + 2$

$a_{1} = 3$

$a_{2} = a_{1} + 2 = 3 + 2 = 5$

$a_{3} = a_{a} + 2 = 5 + 2 = 7$

$a_{4} = a_{3} + 2 = 7 + 2 = 9$

$a_{5} = a_{4} + 2 = 9 + 2 = 5$

Explicit formulas of arithmetic sequences

The explicit formula allows you to simply plug in the number of the term you are interested in to get the value of that term, rather than having to write out each step as with recursive formulas. This is the recursive formula for the arithmetic sequence 3, 5, 7 …

$a_{n} = 3 + 2 (n - 1)$

Example 4

So to find the $5^{th}$ term, we need to plug $n = 5$ into the explicit formula.

$a_{5} = 3 + 2 (5 - 1)$

$a_{5} = 3 + 2 \times 4$

$a_{5} = 3 + 8$

$a_{5} = 11$

Arithmetic sequences are functions

The formulas used for arithmetic sequences work like functions-you input $a$ term number $n$ , and the formula outputs the value of that term $a_{n}$ .

Sequences are actually defined as functions. But you should note that n cannot be any real number value. There's no such thing as a negative fifth term, for example, or the ${0.4}^{th}$ term of a sequence.

In fact, the domain of sequences, which is the set of all possible inputs of the function, is the positive integers.

Practice working with arithmetic sequences

a. Find $a_{4}$ in the sequence where $a_{1} = -5$ and $a_{2} = 4$

The first term is -5

$a_{n} = a_{n - 1} + 9$

Add 9 to the previous term

$a_{n} = a_{n - 1} + 9$

$a_{1} = -5$

$a_{2} = a_{1} + 9 = - 5 + 9 = 4$

$a_{3} = a_{2} + 9 = 4 + 9 = 13$

$a_{4} = a_{3} + 9 = 13 + 9 = 22$

So $a_{4} = 22$

b. Find $a_{10}$ in the sequence given by the function $a_{n} = -5 + 9 (n - 1)$

$a_{10} = -5 + 9 (10 - 1)$

$a_{10} = -5 + 9 \times 9$

$a_{10} = -5 + 81$

$a_{10} = 76$

Topics related to the Arithmetic Sequences

Arithmetic Series

Common Difference

Sum of the First n Terms of an Arithmetic Sequence

Flashcards covering the Arithmetic Sequences

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Arithmetic Sequences

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

Get help learning about arithmetic sequences

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