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Question

D’Andre has decided that he needs to get more exercise, so he is building a running regimen for himself. He decides that, beginning on Sunday, he’ll run 10 feet. On each subsequent day, he’ll double the distance he ran the previous day. So he’ll run 20 feet on Monday, 40 feet on Tuesday, and so on. His Track coach (who also happens to be his math teacher) told D’Andre that he won’t be able to keep to his plan for very long, but D’Andre is determined to make it work for at least two weeks.

a. According to his plan, how far will D’Andre have to run on the 14th day? Give your answer to the nearest tenth of a mile (1 mile = 5,280 feet).

b. If D’Andre sticks to his plan, ho

Answer 1

a. To find the distance D'Andre will have to run on the 14th day, we can use the formula for a geometric sequence:

\[a_n = a_1 \cdot r^{n-1}\]

Where:
- \(a_n\) is the distance on the nth day
- \(a_1\) is the distance on the first day (10 feet)
- \(r\) is the common ratio (2)
- \(n\) is the number of days

Plugging in the values:
\[a_{14} = 10 \cdot 2^{14-1} = 10 \cdot 2^{13} = 10 \cdot 8192 = 81920\text{ feet}\]

Converting to miles:
\[81920\text{ feet} / 5280\text{ feet/mile} = 15.5\text{ miles}\]

Therefore, D'Andre will have to run approximately 15.5 miles on the 14th day.

b. To find the total distance D'Andre will have run over the course of two weeks, we can calculate the sum of a geometric series:

\[S_n = \frac{a_1(r^n-1)}{r-1}\]

Where:
- \(S_n\) is the sum of the first \(n\) terms
- \(a_1\) is the distance on the first day (10 feet)
- \(r\) is the common ratio (2)
- \(n\) is the number of days (14 in this case)

Plugging in the values:
\[S_{14} = \frac{10(2^{14}-1)}{2-1} = \frac{10(16384-1)}{1} = \frac{10(16383)}{1} = 163830\text{ feet}\]

Converting to miles:
\[163830\text{ feet} / 5280\text{ feet/mile} = 31\text{ miles}\]

Therefore, if D'Andre sticks to his plan, he will have run a total of approximately 31 miles over the course of two weeks.