# math

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Rodney and James are each 30 feet from a door. Rodney moved one foot closer and James moved half the distant between himself and the door. If each person continued to move in this fashion, who would be closer to the door after 10 moves? Who reaches the door first?

• math -

In 10 moves, Rodney has covered 10 feet , so he is 20 ft from the door
in 10 moves , James has moved:
15 + 15/2 + 15/4 + 15/8 + 15/16 + .. + 15/2^9
this is a geometric series with a= 15, r = 1/2
sum(10) = 15(1 - (1/2)^10)/(1 - 1/2)
= (1023/1024)(30)
so he is (1/1024)(30) feet away from the door
notice on the 10th move, he is 1/2^10 of the distance away.

moving in this fashion , James will never get to the door
e.g., after 1000 moves he would still be (1/2^1000) of the distance away from the door
whereas Rodney will clearly reach the door after 30 moves.

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