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SOLUTION TO BRAINTEASER NO. 6 DISCLAIMER This solution is but one of many possible methods for solving this problem. Various other solutions may lead to the same answers. This specific solution was provided by the Institute of Mathematical Sciences and Physics of the University of the Philippines Los Baños.
PROBLEM:
A sequence of ants walks from (0,0) to (1,0). The nth ant walks along n semicircles of 1
radius n with diameters lying along the line from (0,0) to (1,0). Let Ln be the length of the path walked by the nth ant. Find lim 𝐿𝑛 . 𝑛→∞
SOLUTION: Recall that the length of a semicircle is equal to half the circumference of the circle, that is 1
1
½(2πr) or simply πr. So at every semicircle, the nth ant walks π(𝑛) units. Thus, Ln = nπ(𝑛). Simplifying that, we obtain Ln = π. ∴ lim 𝐿𝑛 = lim 𝜋 𝑛→∞
ANSWER:
𝑛→∞
π
UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños
