Consider finite iterations of the "snowflake" curve, which is the limit of the curve generated by starting with an equilateral triangle of perimeter 3 and iteratively replacing the middle third of each line segment with two line segments that form 120 degree angles with respect to the original segment, so that the newly added segments and the one being removed form an equilateral triangle. Then each new segment is equal in length to one third of the length of the original line segment. The initial shape is a triangle whose perimeter has length 3. After one step of the iterative process the shape is a six- pointed star with a perimeter of length 4. In general, after n steps are performed the perimeter has length 3*(4/3)" which goes to infinity exponentially fast as n goes to infinity. In an effort to make a "modified snowflake" curve with finite length, even as n goes to infinity, one may consider adjusting the proportion r, of each side length that is replaced in the n" step. For example, if one uses ri = 1/3 and r2 = 1/4, ra = 1/5, r4 = 1/6, etc..., one would replace the middle thirds of each side of the initial triangle to obtain a "star-shaped" figure with 12 sides of length 1/3. However, in the second step one would replace the middle quarter (r2 = 1/4) of each of these 12 sides with two sides that form 120 degree angles with respect to the original segment, so that newly added segments and the one being removed form an equilateral triangle. Then each new segment is equal in length to one quarter of the length of the original line segment. Notice that after this second step you have two different lengths of sides and this situation will become more complicated as you continue, with the curve in the n" step having many different edge lengths. (a) Consider the sequence r, = 1/(n+2). Determine a formula for the perimeter of the n" step curve using this sequence of proportions. Does the sequence of perimeters converge to infinity or to some finite value? (b) Now consider the sequence In = 1/2". For this new choice of ro determine a formula for the perimeter of the n" step curve using this sequence of proportions. Does the sequence of perimeters converge to infinity or to some finite value? (c) Find the limiting area of the inside of the "modified snowflake" curve in both cases (a) and (b)