4 Construct a sequence of figures T T1 T2 T1 see below in th

4. Construct a sequence of figures T T1, T2, T1, (see below) in the following way To is equilateral triangle of side length one TI is obtained from To by replacing the middle third each edge of To by an outward equilateral triangle (of side length T2 is obtained from Ti by replacing the middle third of each edge of Ti by an outward equilateral triangle (of side length 1); and so on. (The area of an equilateral triangle with side length s is s\')

Solution

Area of Triangle To = (sqrt3)/4 * (side)^2 = sqrt3/4

Area of Figure T1 = Area of Triangle To + Area of 3 smaller triangles with length 1/3 units

=> sqrt(3)/4 + 3 * sqrt(3)/4 * (1/3)^2

=> sqrt(3)/4 + sqrt(3)/12

=> sqrt(3)/3

Similarly if we write the recurrence relation

An = A(n-1) + 1/3 * (4/9)^(n-1)

Hence the area added going from n to n+1 will be

Area added = 1/3 * (4/9)^n units

e) Area of Tn

An = A(n-1) + 1/3 * (4/9)^(n-1)

Using Ao = sqrt(3)/4, which is the area of the first triangle To we get

An = 1/5 * ( 8 - 3(4/9)^n) * sqrt(3)/4

f) Limit of An when n tending to infinit

A(infinity) = 1/5 * 8 * sqrt(3)/4

=> 2 * sqrt(3)/5

=> 0.8660 units