9.4 Series and their notations (Page 2/18)

College algebra

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Evaluate $\sum_{k = 2}^{5} (3 k - 1) .$

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} + (a_{1} + d) + (a_{1} + 2 d) + ... + (a_{n} - d) + a_{n} .

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

S_{n} = a_{n} + (a_{n} - d) + (a_{n} - 2 d) + ... + (a_{1} + d) + 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} S_{n} = a_{1} + (a_{1} + d) + (a_{1} + 2 d) + ... + (a_{n} - d) + a_{n} \\ + S_{n} = a_{n} + (a_{n} - d) + (a_{n} - 2 d) + ... + (a_{1} + d) + a_{1} \end{array}}{2 S_{n} = (a_{1} + a_{n}) + (a_{1} + a_{n}) + ... + (a_{1} + a_{n})}

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

2 S_{n} = n (a_{1} + a_{n}) .

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

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

A General Note

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 (a_{1} + a_{n})}{2}

How To

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

Identify $a_{1}$ and $a_{n} .$
Determine $n .$
Substitute values for $a_{1}, a_{n},$ and $n$ into the formula $S_{n} = \frac{n (a_{1} + a_{n})}{2} .$
Simplify to find $S_{n} .$

Finding the first n Terms of an arithmetic series

Find the sum of each arithmetic series.

$5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32$
$20 + 15 + 10 +…+ −50$
$\sum_{k = 1}^{12} 3 k - 8$

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.

$\begin{array}{l} \begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \end{array} \\ S_{10} = \frac{10 (5 + 32)}{2} = 185 \end{array}$
We are given $a_{1} = 20$ and $a_{n} = - 50.$

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

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

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

$\begin{array}{l} \begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \end{array} \\ S_{15} = \frac{15 (20 - 50)}{2} = - 225 \end{array}$
To find $a_{1},$ substitute $k = 1$ into the given explicit formula.

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

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

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

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

$\begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \\ S_{12} = \frac{12 (- 5 + 28)}{2} = 138 \end{array}$

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

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$1 .4 + 1 .6 + 1 .8 + 2 .0 + 2 .2 + 2 .4 + 2 .6 + 2 .8 + 3 .0 + 3 .2 + 3 .4$

$26 .4$

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$13 + 21 + 29 + \dots + 69$

$328$

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

$−280$

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

We can now use the formula for arithmetic series.

\begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \\ S_{8} = \frac{8 (\frac{1}{2} + \frac{9}{4})}{2} = 11 \end{array}

She will have walked a total of 11 miles.

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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