A house of cards, a structure created by stacking playing cards on top of each other in the shape of a pyramid. Each small triangle is made using \(3\) cards and each layer has \(1\) less triangle than the layer below it. Two friends Vinoth and Vikranth making fun of this structure.
(i) Vinoth and Vikranth want to use \(3\) cards in the top layer and \(18\) in the bottom layer. Form an A.P showing the number of cards in each layer starting from the top layer.
(ii) Suppose if the pyramind has \(10\) layers, determine the number of cards in the \(10^{th}\) layer.
(iii) Vinoth is planning to make a pyramid with the top and bottom layer containing \(15\) and \(138\) cards respectively. How many layers will such a pyramid have?
(iv) If they have \(360\) cards. Find the maximum number of layers they can make a triangle with \(3\) cards at the top layer.