How many squares are in the first eight figures of the Sierpinski carpet?

Each new figure in the Sierpinski carpet contains 8 copies of each square from the previous figure, so the numbers of squares are 1, 8, 64, and so on. The total in the first eight figures is 1 + 8 + 8² + … + 8⁷ = (8⁸ − 1) ÷ 7 = 2,396,745. Thus, the answer is 2,396,745 squares.

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