Arithmetic sequences exercises can be solved using the arithmetic sequence formula. This formula allows us to find any number in the sequence if we know the common difference, the first term, and the position of the number that we want to find.

Here, we will look at a summary of arithmetic sequences. In addition, we will explore several examples with answers to understand the application of the arithmetic sequence formula.

## Summary of arithmetic sequences

An arithmetic sequence is a list of numbers that has a defined pattern. We can determine if a sequence is arithmetic by taking any number and subtracting it by the previous number. Arithmetic sequences have a constant difference between consecutive numbers.

The constant difference between the consecutive numbers of an arithmetic sequence is called the **common difference** and denoted by the letter *d*. If the common difference is positive, we have an increasing arithmetic sequence and if the common difference is negative, we have a decreasing arithmetic sequence:

We can find different terms of the arithmetic sequence using the following formula:

To determine any term in the arithmetic sequence, we have to know the common difference, a term in the sequence, and the position of the term that we want to determine.

## Arithmetic sequences – Examples with answers

Using the formula detailed above, we can solve various arithmetic sequence examples. Each of the following examples has its respective detailed solution. Try to solve the problems yourself before looking at the solution so that you can practice and fully master this topic.

**EXAMPLE 1**

Find the next term in the arithmetic sequence: 3, 7, 11, 15, **?**.

##### Solution

First, we have to find the common difference of each pair of consecutive numbers:

- $latex 15-11=4$
- $latex 11-7=4$
- $latex 7-3=4$

The common difference is 4. To find the next term after 15, we simply have to add 4 to 15. Therefore, we have $latex 15+4=19$.

**EXAMPLE 2**

Find the next term in the sequence: 28, 23, 18, 13, **?**.

##### Solution

We start by finding the common difference:

- $latex 13-18=-5$
- $latex 18-23=-5$
- $latex 23-28=-5$

In this case, we obtained a negative common difference. To find the next term, we just have to add this difference to the last term. Thus, we have $latex 13+(-5)=8$.

**EXAMPLE 3**

Find the following two terms in the arithmetic sequence: -17, -13, -9, -5, **?**, **?**.

##### Solution

At first glance, we may think that we have a negative common difference since we have negative numbers, but we have to remember that when the sequence is growing, the common difference is positive:

- $latex -5-(-9)=4$
- $latex -9-(-13)=4$
- $latex -13-(-17)=4$

We see that the common difference is positive 4 because the arithmetic sequence is growing. We can obtain the following two terms by adding the common difference to the last term:

$latex -5+4=-1$

$latex -1+4=3$

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**EXAMPLE 4**

Find the term 26 in the arithmetic sequence: 3, 6, 9, 12, …

##### Solution

In this case, we have to use the arithmetic sequence formula $latex a_{n} = a_{1} + (n-1) d$. To use this formula, we have to know the first term, the common difference, and the position of the term we want to find:

- First term: $latex a_{1}=3$
- Common difference: $latex d=3$
- Position of term: $latex n=26$

Now, we substitute these values in the formula and solve:

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

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

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

$latex a_{26}=3+75$

$latex a_{26}=78$

**EXAMPLE 5**

Find the term 22 in the arithmetic sequence: 15, 8, 1, -6, …

##### Solution

We have to find the first term, the common difference and the position of the term to substitute in the arithmetic sequence formula:

- First term: $latex a_{1}=15$
- Common difference: $latex d=-7$
- Position of terms: $latex n=22$

We substitute these values in the formula:

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

$latex a_{22}=15+(22-1)(-7)$

$latex a_{22}=15+(21)(-7)$

$latex a_{22}=15-147$

$latex a_{22}=-132$

**EXAMPLE 6**

Find the term 16 in the arithmetic sequence: $latex \frac{5}{2}$, 3, $latex \frac{7}{2}$, 4, …

##### Solution

In this case, we have fractional numbers, but similar to the previous problems, we just have to find the different values to substitute in the arithmetic sequence formula:

- First term: $latex \frac{5}{2}$
- Common difference: $latex \frac{1}{2}$
- Position of term: $latex n=16$

Now, we use the formula with these values:

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

$latex a_{16}=\frac{5}{2}+(16-1)\frac{1}{2}$

$latex a_{16}=\frac{5}{2}+(15)\frac{1}{2}$

$latex a_{16}=\frac{5}{2}+\frac{15}{2}$

$latex a_{16}=10$

## Arithmetic sequences – Practice problems

Solve the following arithmetic sequences problems and test your knowledge on this topic. Use the arithmetic sequence formula detailed above to solve the exercises. If you have problems with these exercises, you can study the examples solved above.

## See also

Interested in learning more about sequences? Take a look at these pages:

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