Discovering Fibonacci relationships (S)
We'll use the symbol F1 to stand for the first Fibonacci number, F2 for the second Fibonacci number, F3 for the third Fibonacci number, and so forth. So F1 = 1, and F2 = 1, and, therefore, F3 = F2 + F1 = 2, and F4 = F3 + F2 = 3, and so on. In other words, we write Fn for the nth Fibonacci number where n represents any natural number; for example, we denote the 10th Fibonacci number as F10 and hence
we have F10 = 55. So, the rule for generating the next Fibonacci number by adding up the previous two can now be stated, in general, symbolically as:
Fn = Fn − 1 + Fn − 2
By experimenting with numerous examples in search of a pattern, determine a simple formula for (Fn +1)2 + (Fn)2; that is, a formula for the sum of the squares of two consecutive Fibonacci numbers.