Consider first the horizontal movement of the bug. The bug moves horizontally on each odd-numbered move, with the horizontal movements alternating positive and negative directions. On the first move, the bug moves 1 unit horizontally, on the third -1/4 horizontally, on the fifth 1/16 horizontally, on the seventh -1/64 horizontally, and so on.
Combining successive positive and negative movements, we have movement of 3/4, and then 3/64, and so on. This is an infinite geometric sequence with initial term 3/4 and common ration 1/16. Its sum is thus (3/4)/(1 - 1/16) = (3/4)/(15/16) = 4/5.
Next consider the vertical movement of the bug. The bug moves vertically on each even-numbered move, with the vertical movement alternating positive and negative directions. On the second move, the bug moves 1/2 unit vertically, on the fourth -1/8, on the sixth 1/32, on the eighth -1/128, and so on.
Combining successive positive and negative movements, we have movements of 3/8, and then 3/128, and so on. This is an infinite geometric sequence with initial terms 3/8 and common ratio 1/16. Its sum is thus (3/8)/(1 - 1/16) = (3/8)/(15/16) = 2/5.
In the limit, then, the bug approaches the point (4/5, 2/5).
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