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Chapter 7: Problem 17

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 50 days?

Short Answer

Expert verified
After 50 days a total of 806.25 miles is ridden.

Step by step solution

01

Identify the arithmetic sequence and its terms

In this case, the arithmetic sequence is the series of distances ridden each day. The first term (a) is 10 miles, and the common difference (d) between consecutive terms is 0.25 miles. Following the formula for an arithmetic sequence, we can find the n-th term (a_n) for any day n as: \[a_n = a + (n - 1)d\]
02

Find the 50th term of the arithmetic sequence

We are given the total number of days to be considered as 50. We need to find the distance ridden on the 50th day to help us calculate the total distance ridden. Based on the formula identified in Step 1, we can find the 50th term by plugging n = 50, a = 10, and d = 0.25: \[a_{50} = 10 + (50-1)(0.25) = 10 + 49(0.25)\] Calculating the value, \[a_{50} = 10 + 12.25 = 22.25\] So, on the 50th day the distance ridden is 22.25 miles.
03

Calculate the total miles ridden

Now, we want to find the sum of the arithmetic series for the total distance ridden for 50 days. To complete this task, we will use the following arithmetic series sum formula: \[S_n = \frac{n}{2}(a + a_n)\] Where S_n is the sum of the n terms of the arithmetic sequence (Riding distance for 50 days), n is the number of terms (50 days), a is the first term (10 miles) and a_n is the n-th term (50th term in this case, 22.25 miles). Plugging in the values, we have: \[S_{50} = \frac{50}{2}(10 + 22.25) = 25(32.25)\] Finally, we calculate the total miles ridden after 50 days, \[S_{50} = 25(32.25) = 806.25\] So, after 50 days a total of 806.25 miles is ridden.

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