



 
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 $75 (n1) = 75  n+1 = 76n$ miles per hour for 1 mile so the time taken is \[\frac{1}{76n} \mbox{hours.}\] In the last mile, the $75^{th}$ mile the time taken is \[\frac{1}{7675} = 1 \mbox{ hour.}\] 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  


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