Fractal Cubes

The image above shows a construction of fractals by joining smaller and smaller cubes to each face of one single cube.

  • We start with a cube of side length 3.
  • In the second figure, the 6 new cubes are formed with a side length of \(\dfrac{1}{3}\) of its previous cube.
  • Then from the third figure and so on, the new cubes can only be formed on the 5 faces of the previous cubes with side length \(\dfrac{1}{3}\) of its previous cube.

This recursion continues indefinitely.

If \(a_{n}\) is the total surface area of the \(n^\text{th}\) figure (from the left), what is the value of \(\displaystyle \lim_{n \to \infty} a_{n}\)?


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