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algebra intermediate

Problem

An equilateral triangle is drawn with a side of length . A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is:
(A)
(B)
(C)
(D)
Solution
The perimeter of the first triangle is . The perimeter of the 2nd triangle is half of that, after drawing a picture. The 3rd triangle's perimeter is half the second's, and so on. Therefore, we are computing . The starting term is , and the common ratio is . Therefore, the sum is
Final answer
D