Exercise 13.

Consider the sequence of partial sums of squares of Fibonacci numbers: \(F_1^2\text{,}\) \(F_1^2 + F_2^2\text{,}\) \(F_1^2 + F_2^2 + F_3^2, \ldots\text{.}\) The sequences starts \(1, 2, 6, 15, 40,\ldots\)

  1. Guess a formula for the \(n\)th partial sum, in terms of Fibonacci numbers. Hint: write each term as a product.

  2. Prove your formula is correct by mathematical induction.

  3. Explain what this problem has to do with the following picture:

in-context