# 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.

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