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Chapter 7: Problem 18

Suppose you started an exercise program by riding your bicycle 10 miles on the first day and then you increased the distance you rode by 0.25 miles each day. How many total miles did you ride after 70 days?

Short Answer

Expert verified
After 70 days of the exercise program, you would have ridden a total of 1306.25 miles.

Step by step solution

01

1. Identify the formula for arithmetic series

We can find the sum of an arithmetic series using the formula: \(S_n = \frac{n(a_1 + a_n)}{2}\), where S_n is the sum of n terms, n is the number of terms (days), a_1 is the first term (distance on the first day), and a_n is the nth term (distance on the nth day).
02

2. Identify the given values

In this problem, we are given: - \(a_1\) (first term) = 10 miles - Common difference (d) = 0.25 miles - Number of terms (n) = 70 days
03

3. Find the nth term

Using the formula for the nth term of an arithmetic sequence, we can find the 70th term: \(a_n = a_1 + (n - 1)d\) Plug in the given values: \(a_{70} = 10 + (70 - 1)(0.25)\) \(a_{70} = 10 + (69)(0.25)\) \(a_{70} = 10 + 17.25\) \(a_{70} = 27.25\) miles
04

4. Find the sum of the series

Now, we can find the sum of the distances using the sum of the arithmetic series formula: \(S_{70} = \frac{70(10 + 27.25)}{2}\) \(S_{70} = \frac{70(37.25)}{2}\) \(S_{70} = \frac{2612.5}{2}\) \(S_{70} = 1306.25\)
05

5. Interpret the result

After 70 days of the exercise program, you would have ridden a total of 1306.25 miles.

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