MA001: College Algebra

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference, $d$. The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first $n$ terms of an arithmetic series as:

$S_{n}=a_{1}+\left(a_{1}+d\right)+\left(a_{1}+2 d\right)+\ldots+\left(a_{n}-d\right)+a_{n}$

We can also reverse the order of the terms and write the sum as

$S_{n}=a_{n}+\left(a_{n}-d\right)+\left(a_{n}-2 d\right)+\ldots+\left(a_{1}+d\right)+a_{1}$

If we add these two expressions for the sum of the first $n$ terms of an arithmetic series, we can derive a formula for the sum of the first $n$ terms of any arithmetic series.

$\begin{gathered} \frac{S_{n}=a_{1}+\left(a_{1}+d\right)+\left(a_{1}+2 d\right)+\ldots+\left(a_{n}-d\right)+a_{n} \\ +\quad S_{n}=a_{n}+\left(a_{n}-d\right)+\left(a_{n}-2 d\right)+\ldots+\left(a_{1}+d\right)+a_{1}} {2 S_{n}=\left(a_{1}+a_{n}\right)+\left(a_{1}+a_{n}\right)+\ldots+\left(a_{1}+a_{n}\right)} \end{gathered}$

Because there are $n$ terms in the series, we can simplify this sum to

$2 S_{n}=n\left(a_{1}+a_{n}\right)$

We divide by 2 to find the formula for the sum of the first $n$ terms of an arithmetic series.

$S_{n}=\frac{n\left(a_{1}+a_{n}\right)}{2}$

Formula for the Sum of the First $n$ Terms of an Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first $n$ terms of an arithmetic sequence is

$S_n=\frac{n(a1+an)}{2}$

How To

Given terms of an arithmetic series, find the sum of the first $n$ terms.

1. Identify $a_1$ and $a_n$.
   
2. Determine $n$.
   
3. Substitute values for $a_1, a_n$, and $n$ into the formula $S_n=\frac{n(a1+an)}{2}$.
   
4. Simplify to find $S_n$.
   

Example 2

Finding the First $n$ Terms of an Arithmetic Series

Find the sum of each arithmetic series.

ⓑ $20 + 15 + 10 +…+ −50$

ⓒ $\displaystyle\sum_{k=1}^{12} 3k-8$

Solution

ⓐ  We are given $a_1=5$ and $a_n=32$.

Count the number of terms in the sequence to find $n=10$.

Substitute values for $a_1,a_n$, and $n$ into the formula and simplify.

$\begin{aligned}S_{n} &=\frac{n\left(a_{1}+a_{n}\right)}{2} \\S_{10} &=\frac{10(5+32)}{2}=185\end{aligned}$

ⓑ  We are given $a_1=20$ and $a_n=−50$.

Use the formula for the general term of an arithmetic sequence to find $n$.

$\begin{aligned} a_{n} &=a_{1}+(n-1) d \\ -50 &=20+(n-1)(-5) \\ -70 &=(n-1)(-5) \\ 14 &=n-1 \\ 15 &=n \end{aligned}$

$\begin{aligned}&S_{n}=\frac{n\left(a_{1}+a_{n}\right)}{2} \\&S_{15}=\frac{15(20-50)}{2}=-225\end{aligned}$

ⓒ  To find $a_1$, substitute $k=1$ into the given explicit formula.

$\begin{aligned}&a_{k}=3 k-8 \\&a_{1}=3(1)-8=-5\end{aligned}$

We are given that $n=12$. To find $a_{12}$, substitute $k=12$ into the given explicit formula.

$\begin{aligned} &a_{k}=3 k-8 \\ &a_{12}=3(12)-8=28 \end{aligned}$

$\begin{gathered}S_{n}=\frac{n\left(a_{1}+a_{R}\right)}{2} \\S_{12}=\frac{12(-5+28)}{2}=138\end{gathered}$

Use the formula to find the sum of each arithmetic series.

Try It #2

$1.4 + 1.6 + 1.8 + 2.0 + 2.2 + 2.4 + 2.6 + 2.8 + 3.0 + 3.2 + 3.4$

Try It #3

$13 + 21 + 29 + …+ 69$

Try It #4

$\sum_{k=1}^{10} 5-6k$

Example 3

Solving Application Problems with Arithmetic Series

On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?

Solution

This problem can be modeled by an arithmetic series with $\alpha_{1}=\frac{1}{2}$ and $d=\frac{1}{4}$. We are looking for the total number of miles walked after 8 weeks, so we know that $n=8$, and we are looking for $S_8$. To find $a_{8}$, we can use the explicit formula for an arithmetic sequence.

$\begin{aligned} &a_{n}=a_{1}+d(n-1) \\ &a_{8}=\frac{1}{2}+\frac{1}{4}(8-1)=\frac{9}{4} \end{aligned}$

We can now use the formula for arithmetic series.

$\begin{aligned} &S_{n}=\frac{n\left(a_{1}+a_{n}\right)}{2} \\ &S_{8}=\frac{8\left(\frac{1}{2}+\frac{9}{4}\right)}{2}=11 \end{aligned}$

She will have walked a total of 11 miles.

Try It #5

A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?