to infinity. The sum of this series can be found by the algebraic formula
| a | |
| S = | ———, |
| (1 - r) |
in which a = 10 and r = 1/10. Substituting the value of a and r we have
| 10 | 10 | |||||
| ——— | —— | 100 | ||||
| S = | 1 | = | 9 | = | —— | = 11-1/9. |
| 1 - —— | — | 9 | ||||
| 10 | 10 |
This may be solved more simply as follows: The hound runs 10 times as fast as the fox, hence 10 times the distance the fox runs equals the distance the hound runs. Then 10 times the distance the fox runs, minus once the distance the fox runs, which is 9 times the distance the fox runs, is 10 rods; and once the distance the fox runs is 1/9 of 10 rods, or 10/9 rods; and 10 times the distance the fox runs, or the distance the hound runs, is 10 times 10/9 or 100/9, or 11-1/9 rods.
If through passenger trains, running to and from Philadelphia and San Francisco daily, start at the same hour from each place (difference of longitude not being considered) and take the same time—seven days—for the trip, how many through trains will the Pacific Express, that leaves the San Francisco depot at 9 P. M. Sunday, have met when it reaches the Philadelphia depot?
Answer.—As the Pacific Express starts from San Francisco, a train which left Philadelphia the previous Sunday reaches San Francisco, which is not to be counted as a meeting of trains. There are, however, six other trains on the way which it will meet. Also, a train starts from Philadelphia on the same Sunday as the train starts from San Francisco, another on Monday, another on Tuesday, etc., up to Saturday—that is, seven trains, all of which it meets, making, with the six trains previously started, thirteen trains in all which it meets. A train leaves Philadelphia on Sunday at the same time the Pacific Express reaches there, but this is not counted as a meeting.
A switch siding to a single-track railroad is just long enough to clear a train of eight cars and a locomotive. How can two trains of sixteen cars and a locomotive, each going in opposite directions, pass each other at this siding and each locomotive remain with, and have the same relative position to its own train after as before passing?
Answer.—Let one train and its locomotive be denoted by A, and the other train and locomotive by B, and let the track be denoted by a b and the siding by c d, and suppose train A to be going in the direction of a b, and train B in the direction of b a. Then let locomotive B, with eight cars, run out toward a, past c, and back up on the siding with its eight cars; then let train A run out toward b, past c; then let B draw its eight cars on to the main track and run out toward a; then let train A back over toward a, past c, and locomotive A be detached from train A and run over toward b and connect with the eight cars of train B and draw them over past c, and back them up on the siding, and then run off the siding and connect again with its own cars and run on toward b, past c; then let locomotive B back its eight cars and, turning on the siding, connect the two halves of its train and move off past a, the train A moving on at the same time past b.