One side of an equilateral triangle is 24cm. The mid points of its sides are joined to form another triangle. Again , joining the mid points of this triangle formsanother triangle and the process is continued infinitely. Determine the sum of the perimeters of all such triangles.

hi,

for first triangle perimeter = 3 *24 = 72

 for second , ( its sides would be half the first one  )

                   perimeter =  3 * 24 /2  = 72 /2

for third , perimeter =  3 * (24/2 )/2  = 72 / 4

and so on 

so the sum of perimeter will be =  

     72 + 72/2 +72 /4 +72/ 8 + .......................

= 72 ( 1 + 1/2 +1/4 + 1/8 +........................)

 = 72 { 1/( 1-1/2) }                  { sum of ∞ g.p. = a/ 1-r )

= 72 *2

 = 144 cm