The nth term of an arithmetic sequence is found using the value of the common difference, the position of the term, and the value of the first term. We subtract 1 from the term position and multiply by the common difference. Then, we add the value of the first term to find the nth term.

Here, we will solve some examples of the nth term of arithmetic sequences. In addition, you can practice your knowledge of this topic with practice problems.

## 10 Examples of the nth term of arithmetic sequences with answers

If you need a quick revision of the formulas of the nth term of arithmetic sequences, you can explore this article.

**EXAMPLE 1**

Find the value of the 6th term of an arithmetic sequence that has an initial value of 2 and a common difference of 3.

##### Solution

In this example, we know the values of the initial value and the common difference directly. Also, we know we have to find the 6th term. Then,

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

Using these values in the formula for the nth term of an arithmetic sequence, we have:

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

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

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

$latex a_{6}=2+15$

$latex a_{6}=17$

**EXAMPLE **2

**EXAMPLE**

What is the value of 8th term of an arithmetic sequence in which the first term is 4 and the common difference is 5?

##### Solution

Similar to the previous example, here we also know the values of the first term and the common difference. Then, we have:

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

By applying these values in the formula of the nth term of an arithmetic sequence, we have:

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

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

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

$latex a_{8}=4+35$

$latex a_{8}=39$

**EXAMPLE **3

**EXAMPLE**

If the first term of an arithmetic sequence is 10 and its common difference is -2, find the value of the 8th term.

##### Solution

Using the given information, we can identify the following values:

- $latex a=10$
- $latex d=-2$
- $latex n=8$

In this case, we have a negative common difference. However, we can use the same formula, since it also applies in these cases:

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

$latex a_{8}=10+(8-1)(-2)$

$latex a_{8}=10+(7)(-2)$

$latex a_{8}=10-14$

$latex a_{8}=-4$

**EXAMPLE **4

**EXAMPLE**

The first four terms of an arithmetic sequence are 3, 6, 9, 12. What is the value of the 11th term of the sequence?

##### Solution

In this example, we know the value of the first term directly, but not the common difference.

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

- $latex a=3$
- $latex d=3$
- $latex n=11$

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

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

$latex a_{11}=3+(11-1)3$

$latex a_{11}=3+(10)3$

$latex a_{11}=3+30$

$latex a_{11}=33$

**EXAMPLE **5

**EXAMPLE**

The first four terms of an arithmetic sequence are 10, 6, 2, -2. Find the 8th term of the sequence.

##### Solution

We start by finding the value of the common difference. Therefore, we have $latex -2-2=-4$. Then, we have the following:

- $latex a=10$
- $latex d=-4$
- $latex n=8$

Applying the formula for the nth term with these values, we have:

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

$latex a_{8}=10+(8-1)(-4)$

$latex a_{8}=10+(7)(-4)$

$latex a_{8}=10-28$

$latex a_{8}=-18$

**EXAMPLE **6

**EXAMPLE**

An arithmetic sequence starts with terms 3, 9, 15, … Find the 9th term of the sequence.

##### Solution

Finding the common difference, we have $latex 15-9=6$. Therefore, we have the following values:

- $latex a=3$
- $latex d=6$
- $latex n=9$

Using these values in the formula for the nth term, we have:

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

$latex a_{9}=3+(9-1)6$

$latex a_{9}=3+(8)6$

$latex a_{9}=3+48$

$latex a_{9}=51$

**EXAMPLE** 7

**EXAMPLE**

Find the 12th term of the arithmetic sequence that starts with the terms 12, 7, 2, …

##### Solution

The common difference value is $latex 2-7=-5$. Thus, we observe the following values:

- $latex a=12$
- $latex d=-5$
- $latex n=12$

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

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

$latex a_{12}=12+(12-1)(-5)$

$latex a_{12}=12+(11)(-5)$

$latex a_{12}=12-55$

$latex a_{12}=-43$

**EXAMPLE **8

**EXAMPLE**

An arithmetic sequence begins with the terms -13, -9, -5, … What is the value of 7th term?

##### Solution

The common difference of the given sequence is $latex -5-(-9)=4$. Then, we have:

- $latex a=-13$
- $latex d=4$
- $latex n=7$

Applying the formula for the nth term with these values, we have:

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

$latex a_{7}=-13+(7-1)4$

$latex a_{7}=-13+(6)4$

$latex a_{7}=-13+24$

$latex a_{7}=11$

**EXAMPLE **9

**EXAMPLE**

An arithmetic sequence has the first four terms: 5, 11, 17, 23. Find the 30th term.

##### Solution

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

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

Using the formula for the nth term with these values, 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 **10

**EXAMPLE**

Find the 26th term of the arithmetic sequence that starts with the numbers 20, 17, 14, …

##### Solution

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

- $latex a=20$
- $latex d=-3$
- $latex n=26$

Using the formula with the given values, we have:

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

$latex a_{26}=20+(26-1)(-3)$

$latex a_{26}=20+(25)(-3)$

$latex a_{26}=20-75$

$latex a_{26}=-55$

## nth term of arithmetic sequences – Practice problems

#### What is the 16th term of an arithmetic sequence that begins with the terms 19, 12, 5, …?

Write the answer in the input box.

## See also

Interested in learning more about arithmetic sequences? You can take a look at these pages:

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