May I invite you to watch this video, which shows that 13 can be written as 2^2 + 3^2, which is the sum of two consecutive Fibonacci numbers squared ('Fibonacci squares').
Q1. Which other terms of the Fibonacci sequence can be written as the sum of two consecutive Fibonacci squares? [why?]
Q2. Show that the rest of the Fibonacci sequence can be written as the difference of two Fibonacci squares... [and deduce that these terms are also multiples of some Fibonacci number (apart from 1)].
Q3. Show that difference of two consecutive Fibonacci squares can be written as the product of the two Fibonacci numbers on either side (of the consecutive ones).
For example, 8^2 - 5^2 = 3 x 13.
Q4. Here's an interesting pattern I discovered after playing around with Cuisenaire rods:
8^2 + 2^2 = 2.(3^2 + 5^2)
13^2 + 3^2 = 2.(5^2 + 8^2)
Can you find any other interesting patterns like this?