# 13.4 Series and their notations  (Page 2/18)

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Evaluate $\sum _{k=2}^{5}\left(3k–1\right).$

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## Using the formula for arithmetic series

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:

${S}_{n}={a}_{1}+\left({a}_{1}+d\right)+\left({a}_{1}+2d\right)+...+\left({a}_{n}–d\right)+{a}_{n}.$

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

${S}_{n}={a}_{n}+\left({a}_{n}–d\right)+\left({a}_{n}–2d\right)+...+\left({a}_{1}+d\right)+{a}_{1}.$

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.

$\frac{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{n}={a}_{1}+\left({a}_{1}+d\right)+\left({a}_{1}+2d\right)+...+\left({a}_{n}–d\right)+{a}_{n}\hfill \\ +\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{n}={a}_{n}+\left({a}_{n}–d\right)+\left({a}_{n}–2d\right)+...+\left({a}_{1}+d\right)+{a}_{1}\hfill \end{array}}{2{S}_{n}=\left({a}_{1}+{a}_{n}\right)+\left({a}_{1}+{a}_{n}\right)+...+\left({a}_{1}+{a}_{n}\right)}$

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

${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$

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 and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ into the formula ${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}.$
4. Simplify to find ${S}_{n}.$

## Finding the first n Terms of an arithmetic series

Find the sum of each arithmetic series.

1. $\sum _{k=1}^{12}3k-8$
1. We are given ${a}_{1}=5$ and $\text{\hspace{0.17em}}{a}_{n}=32.$

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

Substitute values for $\text{\hspace{0.17em}}{a}_{1},{a}_{n}\text{\hspace{0.17em},}$ and $n$ into the formula and simplify.

2. We are given ${a}_{1}=20$ and ${a}_{n}=-50.$

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

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{n}={a}_{1}+\left(n-1\right)d\hfill \\ -50=20+\left(n-1\right)\left(-5\right)\hfill \\ -70=\left(n-1\right)\left(-5\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}14=n-1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}15=n\hfill \end{array}$

Substitute values for ${a}_{1},{a}_{n}\text{,}\text{\hspace{0.17em}}n$ into the formula and simplify.

$\begin{array}{l}\begin{array}{l}\\ {S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}\end{array}\hfill \\ {S}_{15}=\frac{15\left(20-50\right)}{2}=-225\hfill \end{array}$
3. To find ${a}_{1},\text{\hspace{0.17em}}$ substitute $k=1$ into the given explicit formula.

We are given that $n=12.$ To find ${a}_{12},\text{\hspace{0.17em}}$ substitute $k=12$ into the given explicit formula.

Substitute values for ${a}_{1},{a}_{n},$ and $n$ into the formula and simplify.

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

$\text{26}\text{.4}$

$\text{328}$

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

$\text{−280}$

## 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?

This problem can be modeled by an arithmetic series with $\text{\hspace{0.17em}}{a}_{1}=\frac{1}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}d=\frac{1}{4}.\text{\hspace{0.17em}}$ We are looking for the total number of miles walked after 8 weeks, so we know that $n=8\text{,}$ and we are looking for $\text{\hspace{0.17em}}{S}_{8}.\text{\hspace{0.17em}}$ To find ${a}_{8},$ we can use the explicit formula for an arithmetic sequence.

$\begin{array}{l}\begin{array}{l}\\ {a}_{n}={a}_{1}+d\left(n-1\right)\end{array}\hfill \\ {a}_{8}=\frac{1}{2}+\frac{1}{4}\left(8-1\right)=\frac{9}{4}\hfill \end{array}$

We can now use the formula for arithmetic series.

She will have walked a total of 11 miles.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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