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 + Question from Marc: While on a road trip I imagined this random word problem. Suppose I start a trip of 75 miles. My initial speed is 75 miles per hour. After every mile traveled I decrease my speed by one mile per hour. After the first mile I decrease my speed to 74 miles per hour and so on for each subsequent mile traveled. How long will it take to complete the 75 mile journey?

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

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