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?
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.
To solve this problem, we can track the distances of Rodney and James after each move. Let's start with their initial distances from the door, which is 30 feet for both.
After the first move, Rodney moves one foot closer, so his new distance from the door is 30 - 1 = 29 feet.
James, on the other hand, moves half the distance between himself and the door, which is (30 - 0) / 2 = 15 feet. Therefore, James' new distance from the door is 30 - 15 = 15 feet.
After the second move, Rodney moves one foot closer again, so his distance from the door becomes 29 - 1 = 28 feet.
James also moves half the distance between himself and the door, which is (15 - 0) / 2 = 7.5 feet. Since we need to keep track of whole numbers of feet, we can round this down to 7 feet. Therefore, James' new distance from the door is 15 - 7 = 8 feet.
Following the pattern, we can continue tracking the distances after each move:
Rodney:
- Move 3: 28 - 1 = 27
- Move 4: 27 - 1 = 26
- Move 5: 26 - 1 = 25
- Move 6: 25 - 1 = 24
- Move 7: 24 - 1 = 23
- Move 8: 23 - 1 = 22
- Move 9: 22 - 1 = 21
- Move 10: 21 - 1 = 20
James:
- Move 3: 8 - 1 = 7
- Move 4: 7 - 1 = 6
- Move 5: 6 - 1 = 5
- Move 6: 5 - 1 = 4
- Move 7: 4 - 1 = 3
- Move 8: 3 - 1 = 2
- Move 9: 2 - 1 = 1
- Move 10: 1 - 1 = 0
After 10 moves, Rodney is 20 feet from the door, and James has reached the door, so he is 0 feet away. Therefore, James is both closer to the door after 10 moves and reaches the door first.