Every fourth element in the Fibonacci sequence is divisible by 3. This means that for every nnn, the Fibonacci number F4nF_{4n}F4n is a multiple of 3. For example, the sequence starts as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... and the 4th, 8th, 12th terms (3, 21, 144, etc.) are all divisible by 3
. This property can be proved using mathematical induction by showing that if F4nF_{4n}F4n is divisible by 3, then F4(n+1)F_{4(n+1)}F4(n+1) is also divisible by 3, using the recursive definition of Fibonacci numbers
. Thus, the number that every fourth Fibonacci number is divisible by is 3.
