nth Term of an Arithmetic Sequence – Step-by-step

The terms of an arithmetic sequence can be obtained by adding a number called the common difference to the previous term. Considering this, we can get a formula for the general term of an arithmetic progression, which uses the value of an initial term, the common difference, and the position of the required term.

Here, we will look at the formula that we can use to find the general term of an arithmetic sequence. Then, we will apply this formula to solve some examples.

ALGEBRA
Formula for the nth term of an arithmetic sequence

Relevant for

Learning to find the general term of an arithmetic sequence.

See steps

ALGEBRA
Formula for the nth term of an arithmetic sequence

Relevant for

Learning to find the general term of an arithmetic sequence.

See steps

Steps to find the nth term of an arithmetic sequence

Consider the sequence of numbers 1, 3, 5, 7, … Each term in this sequence is obtained by adding 2 to the previous term. This is an example of an arithmetic sequence.

Arithmetic sequences are sequences in which any term is formed from the previous term by adding a certain number called the common difference.

Generally, the terms of an arithmetic sequence can be obtained using the following formula:

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

where,

  • $latex a$ is the first term of the sequence.
  • $latex d$ is the common difference.
  • $latex n $ is the position of the term.
Formula for the nth term of an arithmetic sequence

Then, we can take the following steps to find the general term of an arithmetic sequence.

1. Identify the value of the first term.

2. Find the value of the common difference.

The common difference is found by subtracting a term from the sequence by the previous term.

3. Use the formula for the general term.

Substitute the values of the first term, the common difference, and the position of the term into the formula $latex a_{n}=a+(n-1)d$.


Solved examples of the nth term of an arithmetic sequence

EXAMPLE 1

What is the value of the 6th term of an arithmetic sequence that has an initial term of 3 and a common difference of 2?

In this case, we have the values of the first term and the common difference given directly. Also, we want to find the 6th term, so we have:

  • $latex a=3$
  • $latex d=2$
  • $latex n=6$

Using these values into the formula for the general term and solving, we have:

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

$latex a_{6}=3+(6-1)2$

$latex a_{6}=3+(5)2$

$latex a_{6}=3+10$

$latex a_{6}=13$

EXAMPLE 2

If the first term of an arithmetic sequence is 5 and its common difference is 4, find the value of the 8th term.

Similar to the previous example, we have the values of the first term and the common difference given in the statement. Thus, we have:

  • $latex a=5$
  • $latex d=4$
  • $latex n=8$

Using these values with the formula for the general term, we have:

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

$latex a_{8}=5+(8-1)4$

$latex a_{8}=5+(7)4$

$latex a_{8}=5+28$

$latex a_{8}=33$

EXAMPLE 3

The first four terms of an arithmetic sequence are 3, 6, 9, 12. Find the 10th term of the sequence.

In this case, we know the first four terms, so we know the first term directly.

To find the common difference, we can subtract any term by its previous term. For example, $latex 12-9=3$. Then, we have:

  • $latex a=3$
  • $latex d=3$
  • $latex n=10$

Using the formula for the general term, we have:

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

$latex a_{10}=3+(10-1)3$

$latex a_{10}=3+(9)3$

$latex a_{10}=3+27$

$latex a_{10}=30$

EXAMPLE 4

Find the 10th term of the arithmetic sequence that starts with the terms 9, 5, 1, …

Similar to the previous example, we know the first term, and we have to find the common difference. In this case, we have $latex 1-5=-4$. Then,

  • $latex a=9$
  • $latex d=-4$
  • $latex n=10$

We have a negative common difference, but we simply have to apply the formula for the general term:

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

$latex a_{10}=9+(10-1)(-4)$

$latex a_{10}=9+(9)(-4)$

$latex a_{10}=9-36$

$latex a_{10}=-27$

EXAMPLE 5

The first four terms of an arithmetic sequence are 5, 11, 17, 23. Find the 30th term.

The common difference of the given sequence is $latex 23-17=6$. Therefore, we have the following values:

  • $latex a=5$
  • $latex d=6$
  • $latex n=30$

Applying the formula, we have:

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

$latex a_{30}=5+(30-1)6$

$latex a_{30}=5+(29)6$

$latex a_{30}=5+174$

$latex a_{30}=179$

EXAMPLE 6

Find the 16th term of the arithmetic sequence starting with the numbers 20, 17, 14, …

The common difference of the given sequence is $latex 14-17=-3$. Therefore, we have the following values:

  • $latex a=20$
  • $latex d=-3$
  • $latex n=16$

Applying the formula, we have:

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

$latex a_{16}=20+(16-1)(-3)$

$latex a_{16}=20+(15)(-3)$

$latex a_{16}=20-45$

$latex a_{16}=-25$

You can explore additional examples of the nth term of an arithmetic sequence in this article: nth Term of an Arithmetic Sequence – Examples and Practice.


nth term of arithmetic sequences – Practice problems

General term arithmetic sequences quiz
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Find the 12th term of an arithmetic sequence that begins with the terms 15, 11, 7, …

Write the answer in the input box.

$latex a_{12}=$

See also

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