The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, … (each number is the sum of the previous two numbers in the sequence and the first two numbers are both 1).
The sums of the squares of some consecutive Fibonacci numbers are given below:
Is the sum of the squares of consecutive Fibonacci numbers always a Fibonacci number?
Reveal Solution
The nth term of the Fibonacci sequence is:
The sequence of two consecutive terms can be written as:
Adding the squares of two consecutive terms gives:
Which is a Fibonacci number.