Question

An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues infinitely. Find the sum of the perimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.

• A. 288 units
• B. 72 units
• C. 36 units
• D. 144 units

Answer 1

The correct option is D 144 units

The side of the first equilateral triangle being 24 units, the first perimeter is 72 units. The second perimeter would be half of that and so on hence forming a GP as 72, 36, 18... with first term = 72 and common ratio(r) = $\frac{1}{2}$.
$\therefore$ Sum of infinite GP = $\frac{72}{1-\frac{1}{2}}=144.$