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A man moves 1km east, 2km north, 3km west, 4km south, 5km east, 6km north, 7km west and so on until he travels total of 300km. So what will be the distance from origin?

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up vote 4 down vote accepted

To travel 300 km, he would need to take 24 "steps".

For every 4 steps, he travels $2\sqrt{2}$ km south west. He will do this 6 times, so the answer is $12\sqrt{2}$ km

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As $1+2+ \cdots +n = \frac{n}{2} (n+1) $, and $300 = 12 \times 25 $, the final movement after 300km is 24km. This is in the direction south, which is easily deduced.

The total distance moved north is $2 + 6 + \cdots + 22$. This is a simple arithemetic series, the sum of which is 72.

By the same method, the total distance moved east is 66, south is 84, and west is 78.

Therefore, the total horizontal displacement is 12, and the total vertical displacement is 12.

So, the displacement from the origin = $\sqrt{12^2 + 12^2} =12\sqrt{2} $.

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