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you start with an equilateral triangle and then trisect each of the sides with a new equilateral triangle on the middle third of each trisection, repeat the process indefinitely
how would you prove that the perimeter is infinite?

  • math -

    You start out with three sides of length a, and the figure perimeter is 3a.

    After the first step you described, you have a figure with 12 sides of length a/3. The perimeter is now 4a.

    After a similar second step, you will have 48 sides of length a/9. The perimeter is now 5.33 a

    After N such steps, you have 3*4^N sides of length a/3^N, with a perimeter of = 3a*(4/3)^N

    That number clearly becomes infinite as N becomes infinite

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