Starting at point P, a fly moves 1 foot east, 2 feet west, 3 feet east, 4 feet west, and continues alternately making east and west moves, each time lengthening the distance by 1 foot.
a. In relation to point P, where is the fly after 50 moves?
b. Where is the fly after 55 moves?
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To answer these questions, we need to consider the pattern in the fly's movement. Let's break down the pattern step by step and analyze how the distances and directions change.
1. The fly's first move is 1 foot east from point P.
2. The second move is 2 feet west from its first position.
3. The third move is 3 feet east from its second position.
4. The fourth move is 4 feet west from its third position.
Analyzing the pattern, we notice that the fly alternates between moving east and west. Moreover, the distance of the fly's movement increases by 1 foot after every two moves.
Let's determine the location of the fly after 50 moves:
First, we need to figure out the number of cycles the fly completes, where a cycle consists of one eastward and one westward move. Each cycle takes 4 moves (1 east, 2 west, 1 east).
To find the number of cycles, we divide 50 by 4:
50 รท 4 = 12 cycles (with a remainder of 2)
This means the fly completes exactly 12 cycles and has 2 moves left. The 12 cycles take the fly away from point P, and the remaining 2 moves will bring it back towards P. Since the fly moves west after an even number of moves, it will end up west of P.
Now let's calculate the distance west of point P the fly ends up after 12 cycles. For each complete cycle, the net distance west of P is 1 foot (2 feet west - 1 foot east = 1 foot).
Therefore, after 12 cycles, the fly is 12 feet west of point P.
Now let's consider the remaining 2 moves. The fly moves 1 foot east and 2 feet west. So, it moves a total of 1 foot west (2 feet west - 1 foot east = 1 foot).
Adding the remaining 1 foot to the already moved 12 feet west, the fly's final position after 50 moves is 13 feet west of point P.
Now let's determine the location of the fly after 55 moves:
Following the same analysis, we discover that after 55 moves, the fly completes 13 cycles (52 moves) and has 3 moves remaining (53rd, 54th, and 55th moves).
At this point, the fly is 13 feet west of point P due to the 13 cycles. The remaining 3 moves include 1 eastward and 2 westward movements, resulting in a net distance of 1 foot west (2 feet west - 1 foot east = 1 foot).
Adding the remaining 1 foot to the 13 feet west, the fly's final position after 55 moves is 14 feet west of point P.