An optical fiber is manufactured by depositing successive cylindrical layers of…
MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series
Problem
An optical fiber is manufactured by depositing successive cylindrical layers of transparent polymer around a central core. The initial layer is 1.0 μm thick. Each new layer has thickness equal to 25\% of the thickness of the layer immediately beneath it. The process could, in principle, continue indefinitely.
(a) Provide a reasoned argument, without performing an explicit numerical summation, that the total possible thickness of the fiber cladding is finite even if layers are added without bound.
(b) Calculate the exact total thickness (in μm) that would be obtained if infinitely many layers were deposited.
(c) In practice, the manufacturer stops after the seventh layer (that is, after the initial layer plus six additional layers). Compute the percentage by which the actual cladding thickness after seven layers falls short of the theoretical maximum found in part (b). Round the percentage to two decimal places.