Proof that Every 3rd Fibonacci Number is Even:

We want to prove that every third Fibonacci number is even. To begin, let's define the Fibonacci sequence:

$F_0 = 0, \quad F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2} \quad \text{for} \quad n \geq 2.$

The goal is to show that $F_3, F_6, F_9, F_{12}, \dots$ are all even numbers. Let’s check the Fibonacci numbers for the first few terms to observe any patterns:

$F_0 = 0, \quad F_1 = 1, \quad F_2 = 1, \quad F_3 = 2, \quad F_4 = 3, \quad F_5 = 5,$ $F_6 = 8, \quad F_7 = 13, \quad F_8 = 21, \quad F_9 = 34, \quad F_{10} = 55, \quad F_{11} = 89, \quad F_{12} = 144.$

Notice that $F_3 = 2, F_6 = 8, F_9 = 34, F_{12} = 144$ are all even numbers. Let’s now examine the parity (evenness or oddness) of the Fibonacci numbers systematically and prove that every third Fibonacci number is even.

Step 1: Analyze the Fibonacci sequence modulo 2

We will analyze the Fibonacci sequence modulo 2, i.e., we will check whether each Fibonacci number is even (0 mod 2) or odd (1 mod 2). This will reveal a repeating pattern in the parity of Fibonacci numbers.

We already know the first few terms of the Fibonacci sequence:

$F_0 = 0, \quad F_1 = 1, \quad F_2 = 1, \quad F_3 = 2 \quad (\text{even}, 0 \mod 2), \quad F_4 = 3 \quad (\text{odd}, 1 \mod 2), \quad F_5 = 5 \quad (\text{odd}, 1 \mod 2),$ $F_6 = 8 \quad (\text{even}, 0 \mod 2), \quad F_7 = 13 \quad (\text{odd}, 1 \mod 2), \quad F_8 = 21 \quad (\text{odd}, 1 \mod 2), \quad F_9 = 34 \quad (\text{even}, 0 \mod 2).$

Now let's continue the Fibonacci sequence modulo 2:

$F_{10} = 55 \quad (\text{odd}, 1 \mod 2), \quad F_{11} = 89 \quad (\text{odd}, 1 \mod 2), \quad F_{12} = 144 \quad (\text{even}, 0 \mod 2).$

So, the sequence of Fibonacci numbers modulo 2 looks like this:

$0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, \dots$

Step 2: Identify the repeating pattern

From the sequence above, we can see that the Fibonacci numbers modulo 2 repeat in a cycle every 3 terms:

$(0, 1, 1).$

Thus, every third Fibonacci number is congruent to 0 modulo 2, meaning that every third Fibonacci number is even.

Conclusion:

We have shown that every third Fibonacci number is even, as the Fibonacci sequence modulo 2 repeats in a cycle of length 3, with every third term being 0 (even). Therefore, the claim is proven.

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Would you like further details on the proof, or do you have any questions?

Here are 5 related questions for further exploration:

1. How does this modulo 2 pattern extend to larger Fibonacci numbers?
2. What other properties of Fibonacci numbers can be explored using modular arithmetic?
3. How can you generalize this result to show that every nth Fibonacci number is divisible by a certain number for certain values of n?
4. What is the closed-form expression for the Fibonacci sequence (Binet's formula) and how does it relate to the parity of Fibonacci numbers?
5. How can you prove the periodicity of Fibonacci numbers modulo any integer $m$?

Tip: When working with Fibonacci numbers, examining them modulo smaller integers like 2, 3, or 5 often reveals interesting patterns that can simplify complex proofs.