Example 3 from Algebra and Trigonometry, 13.4 Series and Their Notations

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?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
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.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

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?

How did it go?

Solution

Your turn

Solution

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