The long corridor which leads to the Piccabonus Bank's main vault consists of two flat vertical walls parallel to each other, the North Wall and the South Wall, which have a mirror finish. A security laser shines its beam from the point \(N_0\) on the North Wall, gets reflected off the South Wall and is detected back at \(N_0\).

An earthquake happens and shifts the South Wall which is still flat and vertical but is no longer parallel to the North Wall. As a result, the laser beam shining from \(N_0\) is now reflected off the South Wall at \(S_1\), then off the North Wall at \(N_1\), then off the South Wall at \(S_2\) and so on (see diagram).

Trying to assess the damage, Engineer Ada uses an interferometer and finds that, since the laser beam leaves \(N_0\), it travels exactly 332.8935m until it hits the North Wall for the 50th time at \(N_{50}\). Then it travels exactly 662.687m between \(N_{50}\) and \(N_{100}\).

Ada needs to know the length, in metres, of the path the laser beam travels between \(N_{100}\) and \(N_{150}\). But due to health and safety concerns, it is impossible to move the equipment to \(N_{150}\). Calculate the theoretical value of the length of that path.

Note: the answer is a finite decimal, and you have to enter all the significant figures.

MathsBombe Competition 2019 is organised by the The School of Mathematics at the
University of Manchester.

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