Question

10.3. IUT 3 QueSLUIT HELP Let lo be an equilateral triangle with sides of length 13. The figure 14 is obtained by replacing the middle third of each 1.3 side of lo by a new outward equilateral triangle with sides of length triangle with sides of length 1 . The process is repeated where In +7 is obta ocess is repeated whe obtained 13 Answer 30 +1. Answ by replacing the middle third of each side of In by a new outward equilateral triangle with sides of length parts (a) and (b). to As ^ y lzs 13 a. Let Pn be the perimeter of In: Show that lim Pn = 0. no Pn= , so lim Pn = 00. no (Type an exact answer.)

Answer 1

It is easy to get that perimeter of triangle equilateral I0 is 39 .

For triangle I1 we replace each of the three sides with 4 segments , so there are total 12 segments for this triangle and each segment has length one third of its previous length so length of new segment is (13/3).

So perimeter of triangle I1 = 12*(13/3)= 52

Similarly for triangle I3 there are total 3*4*4 = 48 line segments and perimeter is 3*4*4 (13/3)^2

Continuing the same pattern we get the perimeter for nth triangle as

Pn= 3 * 4"

P = 3 *

Which is clearly a geometric series with common ratio = 52/3 That means this series is diveregnt .

so answer would be infinity.

P = 3 *

52 lim Pn=3* lim n->003 n->00