Hi Marc,

Nice problem that has a nice solution, sort of!

For the first mile you travel at 75 miles per hour for 1 mile so the time taken is

\[\frac{1}{75} \mbox{hours.}\]

For the second mile you travel at 74 miles per hour for 1 mile so the time taken is

\[\frac{1}{74} \mbox{hours.}\]

For the third mile you travel at 73 miles per hour for 1 mile so the time taken is

\[\frac{1}{73} \mbox{hours.}\]

I am sure you see the pattern, or the $n^{th}$ mile you travel at $75 -(n-1) = 75 - n+1 = 76-n$ miles per hour for 1 mile so the time taken is

\[\frac{1}{76-n} \mbox{hours.}\]

In the last mile, the $75^{th}$ mile the time taken is

\[\frac{1}{76-75} = 1 \mbox{ hour.}\]

Hence the total time taken is

\[\frac{1}{75} + \frac{1}{74} + \cdot \cdot \cdot + 1 \mbox{ hours}\]

which is more commonly written

\[ 1 + \frac12 + \frac13 + \cdot \cdot \cdot + \frac{1}{74} + \frac{1}{75} \mbox{ hours.}\]

The infinite series

\[1 + \frac12 + \frac13 + \cdot \cdot \cdot + \frac{1}{n} + \cdot \cdot \cdot\]

is called the Harmonic Series and the solution to your problem is a partial sum of the Harmonic Series. The reason I said this is only sort of a nice solution is that there is no easy way to compute a partial sum of the Harmonic Series. You can of course add the fractions but for more than a few terms an approximation seems appropriate. There is an approximation formula given on the Mathematics Stack Exchange web site and using their expression I get a total time of approximately 4.90 hours.

Penny