The interesting thing here, is that a \(3\times 3\) square now has area 13. Our goal is the find a formula for the area of a \(n \times n\) (diagonal) square.
Write out the first few terms of the sequence of areas (assume \(a_1 = 1\text{,}\) \(a_2 = 5\text{,}\) etc). 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, referring to the diagonal squares.
Use your results from part (a) to find a closed formula for the sequence. Show your work. Note, while there are lots of ways to find a closed formula here, you should use partial sums specifically.
Find the closed formula in as many other interesting ways as you can.