The sum of an arithmetic sequence can be found using two different formulas, depending on the information available to us. Generally, the essential information is the value of the first term, the number of terms, and the last term or the common difference.

Here, we will solve several examples of the sum of arithmetic sequences. In addition, we will look some practice problems in which you can apply what you have learned.

## 10 Examples of sums of arithmetic sequences with answers

You can look at a revision of the formulas for the sum of arithmetic sequences in this article.

**EXAMPLE 1**

Find the sum of the first 8 terms of an arithmetic sequence, where the first term is 4 and the 8th term is 25.

##### Solution

We can start by writing all the values we know:

- First term: $latex a=4$
- Last term: $latex l=8$
- Number of terms: $latex n=8$

Now, we can use the formula for the sum of arithmetic sequences with the given values:

$$S_{n}=\frac{n}{2}[a+l]$$

$$S_{8}=\frac{8}{2}[4+25]$$

$$S_{8}=4[29]$$

$$S_{8}=116$$

**EXAMPLE **2

**EXAMPLE**

The first term of an arithmetic sequence is 7 and the 15th term is 63. Find the sum of the first 15 terms.

##### Solution

We start by writing all the known values:

- First term: $latex a=7$
- Last term: $latex l=63$
- Number of terms: $latex n=15$

Now, we use the formula for the sum of an arithmetic sequence:

$$S_{n}=\frac{n}{2}[a+l]$$

$$S_{15}=\frac{15}{2}[7+63]$$

$$S_{15}=7.5[70]$$

$$S_{15}=525$$

**EXAMPLE **3

**EXAMPLE**

Find the sum of the first 9 terms of an arithmetic sequence, where the first term is -20 and the 9th term is -44.

##### Solution

We have the following information:

- $latex a=-20$
- $latex l=-44$
- $latex n=9$

Using the formula for the sum with the given information, we have:

$$S_{n}=\frac{n}{2}[a+l]$$

$$S_{9}=\frac{9}{2}[-20-44]$$

$$S_{9}=4.5[-64]$$

$$S_{9}=-288$$

**EXAMPLE **4

**EXAMPLE**

Find the sum of the first 20 terms of an arithmetic sequence starting with 5, 9, 13, 17, …

##### Solution

In this case, we don’t know the 20th term of the sequence. However, we can find the common difference by subtracting a term by its previous term.

Therefore, we have 9-5=4. And we have the following information:

- First term: $latex a=5$
- Common difference: $latex d=4$
- Number of terms: $latex n=20$

Now, we use this information in the second formula for the sum of an arithmetic sequence:

$$S_{n}=\frac{n}{2}[2a+(n-1)d]$$

$$S_{20}=\frac{20}{2}[2(5)+(20-1)4]$$

$$=10[10+(19)4]$$

$latex =10[10+76]$

$latex =10(86)$

$latex S_{20}=860$

**EXAMPLE **5

**EXAMPLE**

An arithmetic sequence starts with the terms 60, 55, 50, … Find the sum of the first 12 terms.

##### Solution

We can find the common difference by subtracting a term by its previous term: 55-60=-5. Then, we have the following:

- $latex a=60$
- $latex d=-5$
- $latex n=12$

Now, we can use the second formula for the sum of an arithmetic sequence with the given information:

$$S_{n}=\frac{n}{2}[2a+(n-1)d]$$

$$S_{12}=\frac{12}{2}[2(60)+(12-1)(-5)]$$

$$=6[120+(11)(-5)]$$

$latex =6[120-55]$

$latex =6(65)$

$latex S_{12}=390$

**EXAMPLE **6

**EXAMPLE**

Find the sum of the first 25 terms of an arithmetic sequence that starts with the terms 9, -1, -11, …

##### Solution

The common difference of the sequence is -1-9=-10. Then, we have the following values:

- $latex a=9$
- $latex d=-10$
- $latex n=25$

Now, we are going to use the second formula for the sum of an arithmetic sequence:

$$S_{n}=\frac{n}{2}[2a+(n-1)d]$$

$$S_{25}=\frac{25}{2}[2(9)+(25-1)(-10)]$$

$$=12.5[18+(24)(-10)]$$

$latex =12.5[18-240]$

$latex =12.5(-222)$

$latex S_{25}=2775$

**EXAMPLE **7

**EXAMPLE**

What is the result of the following sum of the arithmetic sequence?

$$6+8+10+…30$$

##### Solution

We can start by finding the common difference of the sequence: 8-6=2. Then, we have:

- First term: $latex a=6$
- Common difference: $latex d=2$
- Last term: $latex l=30$

We don’t have the number of terms, so we can use the formula for the nth term to find it:

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

$latex 30=6+(n-1)2$

$latex 24=(n-1)2$

$latex 12=n-1$

$latex n=13$

Now that we have all the required information, we can use the formula for the sum of arithmetic sequences:

$$S_{n}=\frac{n}{2}[a+l]$$

$$S_{13}=\frac{13}{2}[6+30]$$

$$S_{13}=\frac{13}{2}[36]$$

$$S_{13}=234$$

**EXAMPLE **8

**EXAMPLE**

Find the sum of the following arithmetic sequence:

$$9+13+17+…+41$$

##### Solution

The common difference is equal to 13-9=4. Then, we have the following values:

- $latex a=9$
- $latex d=4$
- $latex l=41$

Now, we use the formula for the nth term to find the value of *n*:

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

$latex 41=9+(n-1)4$

$latex 32=(n-1)4$

$latex 8=n-1$

$latex n=9$

Now that we have all the required information, we can find the sum of the sequence:

$$S_{n}=\frac{n}{2}[a+l]$$

$$S_{9}=\frac{9}{2}[9+41]$$

$$S_{9}=\frac{9}{2}[50]$$

$$S_{9}=225$$

**EXAMPLE **9

**EXAMPLE**

What is the sum of the following arithmetic sequence?

$$62+60+58+…+38$$

##### Solution

The common difference of the sequence is equal to 60-62=-2. Then, we have the following values:

- $latex a=62$
- $latex d=-2$
- $latex l=38$

Now, let’s use the formula for the nth term to find the value of *n*:

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

$latex 38=62+(n-1)(-2)$

$latex -24=(n-1)(-2)$

$latex 12=n-1$

$latex n=13$

Now, we find the sum with these values:

$$S_{n}=\frac{n}{2}[a+l]$$

$$S_{13}=\frac{13}{2}[62+38]$$

$$S_{13}=\frac{13}{2}[100]$$

$$S_{13}=650$$

**EXAMPLE **10

**EXAMPLE**

If the first term of an arithmetic sequence is 2 and the nth term is 32, find the value of *n* if the sum of the first *n* terms is 357.

##### Solution

We can use the formula for the nth term with $latex a_{n}=32$:

$latex a+(n-1)d=32$

Also, since we know that the first term is 2, we have:

$latex 2+(n-1)d=32$

$latex (n-1)d=30~~[1]$

Now, we use the formula for the sum with the value $latex S_{n}=357$:

$$ \frac{n}{2}[2a+(n-1)d]=357$$

Since we know that $latex a=2$, we have:

$$ \frac{n}{2}[2(2)+(n-1)d]=357$$

$$ \frac{n}{2}[4+(n-1)d]=357~~[2]$$

Substituting equation [1] into equation [2], we have:

$latex n(4+30)=714$

$latex 34n=714$

$latex n=21$

## Sum of arithmetic sequences – Practice problems

#### What is the result of the following sum of an arithmetic sequence? $$1.3+1.6+1.9+…+4.6$$

Write the answer in the input box.

## See also

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

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