Arithmetic Sequences

1. Identifying Arithmetic Sequences

try it

Consider the sequence 4, 7, 10, 13, 16, ….

Can you guess what the next three terms of the sequence are?

Note that each term is 3 more than the previous term. Then, the next three terms are 19, 22, and 25.

In the previous example, notice that each term is 3 more than its previous term.

The sequence above is an example of an arithmetic sequence.

An arithmetic sequence is a sequence in which every term after the first term is obtained by adding some constant to the previous term. This constant is called the common difference, which is denoted d.

The sequence 4, 7, 10, 13, 16, … from above has common difference When taking any two consecutive terms and subtracting the first from the second, the difference is equal to the common difference.

There are two ways to determine if a sequence is arithmetic:

• Find the difference between every pair of consecutive terms and subtract the first from the second. If all the differences are the same, then the sequence is arithmetic. If not, then the sequence is not arithmetic.
• Determine if the same number is added to get from one term to the next. If so, then the sequence is arithmetic. If not, then the sequence is not arithmetic.

EXAMPLE

Consider the sequence 1, 3, 5, 7, ….

It is an arithmetic sequence since every term is 2 more than its preceding term. Therefore, the common difference is

Another way to check this is to take the differences between two consecutive terms:

Since all the differences are the same, the sequence 1, 3, 5, 7, … is arithmetic.

EXAMPLE

The sequence 100, 95, 92, 87, … is not arithmetic because but which means the sequence does not have a common difference.

try it

Consider the following sequences:

• -1, 4, 9, 14, 19, 24, ...
• 2, 4, 8, 16, 32, 64, ...
• 30, 29, 28, 27, …

Determine if each sequence is arithmetic. If it is arithmetic, state its common difference.

The sequence is arithmetic with common difference The sequence is not arithmetic. The sequence is arithmetic with common difference

terms to know

Arithmetic Sequence

A sequence that is obtained by selecting the first term, then adding the same value to get each subsequent term.

Common Difference

The constant difference between any two consecutive terms in an arithmetic sequence.

2. Finding Terms of an Arithmetic Sequence

When given the first term and a common difference, the arithmetic sequence is formed by repeatedly adding the common difference to get all subsequent terms.

EXAMPLE

An arithmetic sequence is to have first term and common difference Write the next three terms of the sequence.

The common difference tells us that we add 9 to each term to get the next term.

Then, the next three terms are:

• 
• 
• 

The arithmetic sequence as a whole is 20, 29, 38, 47, ….

A similar idea can be used to relate two terms of an arithmetic sequence with common difference d.

EXAMPLE

An arithmetic sequence has term and common difference

Find the value of

Consider this picture:

	Starting at add the common difference 6 times to get the value of
	Substitute and
	Simplify.

Thus,

EXAMPLE

An arithmetic sequence has terms and What is the common difference d, and what is the value of

	and are 7 spaces apart in the sequence, therefore their values are 7 common differences apart.
	Substitute and
	Subtract 8 from both sides.
	Solve for d.

Thus, the common difference for the sequence is

Next, find the value of

	Since is known, relate it to which is 17 spaces after in the sequence.
	Substitute and
	Simplify.

Thus,

Note: since is also known, we could have used the equation which gives the same results.

try it

Consider an arithmetic sequence with common difference d in which you are given the value of and you wish to find the value of

Write a₁₂ in terms of a₄ and d.

Given a₄ = 10 and a₁₂ = 34, find the value of d.

Substitute and into the equation then solve for d:

3. Writing the Formula for the nth Term of an Arithmetic Sequence

The recursive formula for an arithmetic sequence with first term is for

It is also possible to find an explicit formula for the nth term, which is always more desirable.

Suppose an arithmetic sequence has first term and we wish to find a formula for

The picture below shows the first six terms of an arithmetic sequence, and how d is added to get each subsequent term.

From the sequence, we have the following relationships between d, and the other terms of the sequence:

Equation	Relationship
	Add d to to get the value of
	Add d twice to to get the value of
	Add d three times to to get the value of
	Add d four times to to get the value of
	Add d five times to to get the value of

This pattern continues.

Notice that the number of times d is added to is one less than the position of the term we seek.

This leads to the following formula, which is the most conventional way to find the nth term of an arithmetic sequence.

formula to know

nth Term of an Arithmetic Sequence

This formula is typically used since the first term of an arithmetic sequence is usually known.

EXAMPLE

Write a formula for the nth term of the arithmetic sequence: 2, 6, 10, 14, 18, 22, …

The sequence has first term and common difference

Then, the nth term of the sequence is which in simplest form is

watch

The steps for writing a formula for the nth term of the sequence 102, 96, 90, 84, … are shown in the following video.

try it

Consider the sequence 50, 47, 44, 41, ….

Write a formula in simplest form for the nth term of the sequence.

Note that each term is 3 less than the term previous to it, so we assume that the sequence is arithmetic. Then, and Then, use the formula

	Substitute known quantities.
	Distribute -3.
	Combine like terms.

The nth term of the sequence, assuming it is arithmetic, is

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.