Final answer:
Every third Fibonacci number is divisible by 3, and this can be proven using mathematical induction. The pattern is established by observations and then confirmed with base cases followed by induction.
Step-by-step explanation:
To determine which Fibonacci numbers are divisible by 3, we can use the principle of mathematical induction. First, we should look at the sequence to spot a pattern. The Fibonacci sequence can be defined as F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for n ≥ 2. Testing the first few terms, we find that F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3, which is divisible by 3, and F(5) = 5. Observing this, we can conjecture that every third Fibonacci number is divisible by 3.
Now, we proceed with induction. Base cases have demonstrated that F(0) and F(4) are divisible by 3. Assuming F(k) is divisible by 3, we want to show that F(k+3) is also divisible by 3. Since F(k+3) = F(k+2) + F(k+1) and we know from the Fibonacci definition that F(k+2) = F(k+1) + F(k), we can substitute to get F(k+3) = 2*F(k+1) + F(k). By the inductive hypothesis, 3 divides F(k), and hence it must divide F(k+3).
This pattern can be confirmed by continuing to test larger numbers in the sequence or by applying more sophisticated mathematical techniques such as modular arithmetic. The property that every third Fibonacci number is divisible by 3 is therefore proven.