Let A B C be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle A B C and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then :

Step-by-step solution

Area of first = 3 a ^2 4 Area of second = 3 a ^2 4 a ^2 4 = 3 a ^2 16 Area of third = 3 a ^2 64 sum of area = 3 a ^2 4 (1+ 1 4 + 1 16 ) aligned & Q = 3 a ^2 4 1 3 4 = a ^2 3 & perimeter of 1^ st =3 a & perimeter of 2^ nd = 3 a 2 & perimeter of 3^ rd = 3 a 4 & P =3 a (1+ 1 2 + 1 4 + ) & P =3 a 2=6 a & a = P 6 & Q = 1 3 P ^2 36 & P ^2=36 3 Q aligned