A train travels at the rate of 10 miles/hr for the first hour of a trip, at 20 miles/hr for the second hour, at 30 miles/hr for the third hour and so on. How many hours will it take the train to complete a 450-mile journey? Assume that the train makes no intermediate stops.
A. 10
B. 9
C. 8.5
D. 8
E. 5

• The speeds at which the train travels in successive hours are the terms of the arithmetic progression 10, 20, 30,... To find the time (say, t hours) taken by the train to complete the 450-mile journey, the sum of this sequence to the appropriate term is considered as follows.
  Time x Speed = Distance = 450 miles.
  1 hour x (10 + 20 + 30 + ... to t terms) = 450
  Dividing by 10, one obtains 1 + 2 + 3 + ... + t = 45
  Now, 1 + 2 + 3 + ... + t = t (t + 1)/2 (a well-known result worth remembering).
  So, t (t + 1) = 90 giving t = 9.
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