An Infinite Decoration
For a 2016 new year's party, the host starts to construct a small fractal shaped as an equilateral triangle with side length 1 foot. Each year to come, he connects the midpoints of the small triangle to form another equilateral triangle inside it to resemble the new year to come. Assuming that this process is infinitely repeated, what is the sum of all of the areas in this infinite decoration in feet.
(Hint: It is not the area of the initial triangle itself.)
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1 solution
Each subsequent triangle has area that is 1/4 of previous triangle ,hence sum = A+A/4+A/16.........=4A/3=4/3(√3/4(1))= √ 3 / 3