# Problem of the Day

Aliens who live 1,000,000 miles from Earth decide to come to our planet for a visit. They program their space shuttle to travel 100,000 miles per hour for the first 500,000 miles, 200,000 miles per hour for the next 250,000 miles, 400,000 miles for the next 125,000 miles and so on, doubling the speed every time half of the remaining distance is covered. If this pattern is maintained until the aliens reach earth, what will their average speed for the trip be in miles per hour?

A. 120,000
B. 125,000
C. 135,000
D. 150,000
E. 162,000

D. 150,000

See the Solution

### Solution

[latexpage]

Average speed is calculated by dividing the total distance by the total time. The total distance is 1,000,000 miles. The first 500,000 miles takes 5 hours because they are traveling at only 100,000 miles per hour. Because $t=\frac{d}{r}$, every time they halve the distance and double their speed, their time gets multiplied by $\frac{1}{4}$. Thus, the total time would be the sum of the infinite geometric series $5+\frac{5}{4}+\frac{5}{16}+…$ where the first term, $a$, is 5, and the common ratio, $r$, is $\frac{1}{4}$. Using the formula for the sum of an infinite geometric series, $S=\frac{a}{1-r}$, we can determine that the total time is $\frac{5}{1-\frac{1}{4}}=\frac{20}{3}$.

Average speed would then be $\frac{1,000,000}{\frac{20}{3}}=150,000$