Exercise 13.

If you have enough toothpicks, you can make a large triangular grid. Below, are the triangular grids of size 1 and of size 2. The size 1 grid requires 3 toothpicks, the size 2 grid requires 9 toothpicks.

Three toothpicks arranged as the sides of an equilateral triangle.
Nine toothpicks arranged into a triangle with two toothpicks forming each edge, and an upside-down triangle in the center.

  1. Let \(t_n\) be the number of toothpicks required to make a size \(n\) triangular grid. Write out the first 5 terms of the sequence \(t_1, t_2, \ldots\text{.}\)

  2. Find a recursive definition for the sequence. Explain why you are correct.

  3. Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct.

  4. Use your results from part (c) to find a closed formula for the sequence. Show your work.

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