Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.0 AIME Find the answer United States

Problem:
A sequence of ants walk from $(0,0)$ to $(1,0)$ in the plane. The $n$ th ant walks along $n$ semicircles of radius $\frac{1}{n}$ with diameters lying along the line from $(0,0)$ to $(1,0)$ . Let $L_{n}$ be the length of the path walked by the $n$ th ant. Compute $\lim_{n \rightarrow \infty} L_{n}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
A semicircle of radius $\frac{1}{n}$ has length $\frac{1}{2} \pi \left(\frac{2}{n}\right) = \frac{\pi}{n}$ , so $n$ such semicircles have total length $\pi$ .

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