A particle moves in a straight line so that its speed is constant for a mile, then changes abruptly for the next mile, is constant for that mile, then changes again---and so on. The speed varies so that for each mile it's inversely proportional to the total integral miles previously traveled.
If the second mile is traversed in 2 hours, how long does the particle take to travel the nth mile, for n=2,3,4,…?
Make a chart along these lines:
Mile 123⋮ Prev. Miles 012⋮ Rate r1r2=1kr3=2k⋮
Here k is a constant of proportionality. We are given that the first mile took two hours to traverse. So in fact,
thus k=1/2. With this value established, we can complete the table:
Mile 1234⋮n Prev. Miles 0123⋮n−1 Rate r1r2=21r3=21/2=41r4=31/2=61⋮rn=n−11/2=2(n−1)1
For that nth mile, d=1=rt=2(n−1)1t, so t=2(n−1).