Answer:
a. geometric series
b. r_n = 100 × (0.75)^n ft
c. 400
Explanation:
a.
Start: 100 ft
After 1 hour: 75% of 100ft = 100 ft × 0.75
After 2 hours: 75% of 100ft × 0.75 = 100 ft × 0.75²
After 3 hours: 75% of 100 ft × 0.75² = 100 ft × 0.75³
Notice what is happening to the radius as the hours pass.
With each passing hour, the radius is 0.75 times the previous radius.
Since each new term is the previous term multiplied by a constant, 0.75, this is a geometric series.
b.
At each hour, multiply 100 ft by 0.75 raised to the number of the hour.
a_n = 100 × (0.75)^n ft
The nth term is 100 times 0.75 to the nth power.
c.
The sum of all the radii is the sum of a series that has 100 as its first term, and each subsequent term is 0.75 times the previous term.
r_n = 100(0.75)^n
Since the constant ratio has an absolute value less than 1, the series has a sum.
S = a_1/(1 - r)
S = sum of infinite series
a_1 = first term
r = constant ratio
S = 100/(1 - 0.75)
S = 100/0.25
S = 400