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 fun 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 his plan for very long, but D' Andre is determined to make it work for at least two weeks.

How far will D' Andre have to run on the 14th day? In nearest tenth of a mile. (1 mile= 5280 feet)

If he sticks to the plan how far will he have ran in those two weeks? to the nearest mile.

Is there an equation for that one?^ or do I just have to add them all up?

Formula: an= 5(2^n)

Day 1: a1=5(2^1)=10
Day 2: a2=5(2^2)=20
Day 3: a3=5(2^3)=40
Day 14:a14=5(a^15)=81,920
He will run 81,920 feet = 15.5 miles

Add all: 10+20+30+...+81,920= 163,830
He will run a total of 163,830 feet = 31 miles

Thumbs up if it helps

To find out how far D'Andre will have to run on the 14th day, we need to determine the distance he runs each day and find the pattern.

Starting from Day 1 with 10 feet, D'Andre doubles the distance each day. So the distance he runs on Day 2 is 10 feet * 2 = 20 feet. On Day 3, it would be 20 feet * 2 = 40 feet. This pattern continues, so on Day n, the distance D'Andre runs will be 10 feet * 2^(n-1).

To find the distance D'Andre will run on the 14th day, we substitute n = 14 in the equation:

Distance on Day 14 = 10 feet * 2^(14-1) = 10 feet * 2^13.

To convert the distance to miles, we divide the result by 5280 (since 1 mile = 5280 feet):

Distance on Day 14 = (10 feet * 2^13) / 5280 miles.

Calculating the result gives us:

Distance on Day 14 = 81920 feet / 5280 miles = 15.5 miles (approximately, rounded to the nearest tenth).

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

To calculate the total distance D'Andre will have run in two weeks, we need to find the sum of the distances from Day 1 to Day 14. Since the distances form a geometric sequence, we can use the formula for the sum of a geometric series:

Sum = a * (r^n - 1) / (r - 1),

where a is the first term (10 feet) and r is the common ratio (2).

Plugging in the values, we get:

Sum = 10 * (2^14 - 1) / (2 - 1).

Calculating the result gives us:

Sum = 10 * (16384 - 1) = 10 * 16383 = 163,830 feet.

To convert the distance to miles, we divide the result by 5280 (since 1 mile = 5280 feet):

Sum = 163,830 feet / 5280 miles = 31 miles (approximately, rounded to the nearest mile).

Therefore, if D'Andre sticks to the plan, he will have run approximately 31 miles in two weeks.

So to summarize, D'Andre will need to run approximately 15.5 miles on the 14th day, and if he sticks to the plan, he will have run approximately 31 miles in two weeks.

Oh, D'Andre, you are quite the determined runner! Let's calculate how far you'll have to run on the 14th day.

On the 14th day, D'Andre will have to run a total of 10 * (2^13) feet.

Now, let me grab my clown calculator and do some math for you. Voila!

On the 14th day, D'Andre will have to run approximately 819,200 feet.

But I hope he has some clown shoes to cover that distance because feet aren't a good unit for measuring running!

To convert it to miles, we divide by 5280 (the number of feet in a mile):

819,200 feet / 5280 = approximately 155 miles.

Wow, D'Andre will be quite the road warrior!

As for calculating how far he'll have run in two weeks, you can definitely use an equation for that. The sum of a geometric series can be used here, where the first term (a) is 10 feet, the ratio (r) is 2, and the number of terms (n) is 14.

The formula for the sum of a geometric series is: S = (a * (1 - r^n)) / (1 - r)

Plugging in the values, we have:
S = (10 * (1 - 2^14)) / (1 - 2)
Simplified:
S = (10 * (1 - 16,384)) / -1
S = 10 * (-16,383) / -1
S = 163,830 feet (approximately)

To convert that to miles, divide by 5280:
163,830 feet / 5280 = approximately 31.1 miles.

So, if D'Andre sticks to his plan for two weeks, he'll have run approximately 31 miles. That's some serious running, D'Andre! Keep up the humor and the pace!

To find out how far D'Andre will have to run on the 14th day, we can use the formula for the sum of a geometric series. The formula is given as:

Sn = a * (r^n - 1) / (r - 1)

Where:
- Sn is the sum of the series up to the nth term
- a is the first term in the series
- r is the common ratio (in this case, 2)
- n is the number of terms

In this case, the first term (a) is 10 feet, the common ratio (r) is 2, and the number of terms (n) is 14. Let's calculate it:

S14 = 10 * (2^14 - 1) / (2 - 1)
S14 = 10 * (16384 - 1) / 1
S14 = 10 * 16383
S14 = 163,830 feet

To convert this to miles, divide by 5280:

163,830 feet / 5280 = 31.06 miles (rounded to the nearest tenth)

Therefore, D'Andre will have to run approximately 31.1 miles on the 14th day.

To find out how far he will have run in two weeks, we need to calculate the sum of distances for the first 14 days (including the 14th day). This can be done by summing the distances from Day 1 to Day 14.

Since the distances are doubling each day, we can use the formula for the sum of a geometric series again. This time, the number of terms (n) is 14, and the first term (a) is 10 feet.

S14 = 10 * (2^14 - 1) / (2 - 1)
S14 = 10 * (16384 - 1) / 1
S14 = 10 * 16383
S14 = 163,830 feet

Converting this to miles:

163,830 feet / 5280 = 31.06 miles (rounded to the nearest mile)

Therefore, if D'Andre sticks to his plan for two weeks, he will have run approximately 31 miles.

To find how far he runs in feet, f, on the given day d, you use the exponential equation f=5*(2^d). In calculus there is the sum function Σ, but there is another way, watch this:

Adding d from 1 to 3:

5*2+
5*2*2+
5*2*2*2=
5*(2^1+2^2+2^3) or 70.
So the simplest equation for the total number of feet after d days is 5(2^d+2^(d-1)+2^(d-2)…2^1)

So your final answer is that he would have to run 16384 feet (~3.1 miles) on day 14, but if he does he will have run a total of 163830 feet, or 31 miles.