Construct a sequence of figures T_0, T_1, T_2, ..(see below) in the following wa
ID: 2877849 • Letter: C
Question
Construct a sequence of figures T_0, T_1, T_2, ..(see below) in the following way, T_0 is an equilateral triangle of side length one; T_1 is obtained from T_0 by replacing the middle third of each edge of T_0 by an outward equilateral triangle (Of side length 1/3): T_2 is obtained from T_1 by replacing the middle third of each edge of T_1 by an outward equilateral triangle (Of side length 1/9); and so on. (The area of an equilateral triangle with side length s is square root 3 s^2/4.) Find S_n, the number of edges in the perimeter of T_n. Find L_n. the length of each edge in the perimeter of T_n Find P_n, the perimeter of T_n. Find lim_n right arrow infinity P_n. How much area is added in going from T_n to T_n+1? Find A_n, the area of T_n. lim_n right arrow infinity A_n.Explanation / Answer
a) As seen in the picture, with each iteration, one edge is turned into 4 edges.
Thus number of edges, Sn in Tn = 3*4n
b) Length of each edge is reduced by three times in each iteration.
Thus length, Ln = 1/3n
c) Perimeter = Sn * Ln = 3*(4/3)n