Distance from origin

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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What if he started at or near one of the poles? –  Harald Hanche-Olsen Oct 17 '12 at 17:37
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. –  Julian Kuelshammer Oct 18 '12 at 7:44

Observe that every new segment is 1km longer than the previous one. So the total distance is basically the sum of the first $n$ integers. Do you know a closed formula for that (Hint: Gauss). That allows you to determine how long was the last segment and it's easy to calculate its direction. But first make a drawing: it'll show you a pattern.