JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Sequences and Series

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 the sum of area of all the triangles formed in this process, then

Answer: (B)

Solution

With initial side

a

, successive triangles (formed by joining midpoints) have sides

a, a /2, a /4, \dots

. Perimeter sum:

P = 3 a (1 + \frac{1}{2} + \frac{1}{4} + \dots) = 3 a \cdot 2 = 6 a

. Area sum:

Q = \frac{3}{4} a^{2} (1 + \frac{1}{4} + \frac{1}{16} + \dots) = \frac{3}{4} a^{2} \cdot \frac{4}{3} = \frac{a ^{2}}{3}

, so

a^{2} = 3 Q

. Then

P^{2} = 36 a^{2} = 363 Q

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Single correct option