5 cm in diameter is 2.5 cm in radius. If you add 12 layers, each 0.05 cm thick, to that, you have a radius of 2.5+ 12(0.05)= 2.5+ 0.6= 3.1 cm. So you have a cylinder with radius 3.1 cm and length 15 cm. Its volume is \(\displaystyle \pi (3.1)^2(15)= 452.86\) cubic centimeters. The volume of the core is \(\displaystyle \pi (3.5)^2(15)= 294.37\) cubic centimeters. The volume of the wire alone is 452.86- 294.37= 158.49 cubic cm.
The wire has radius 0.025 cm so its cross-sectional area is \(\displaystyle \pi (0.025)^2= 0.00196\) square cm so a wire of length "x" would have volume 0.00196x cubic centimeters. Solve 0.00196x= 158.49 for x. That will be in centimeters so you will have to covert to meters.
(I don't see any "arithmetic progression" in this.)
