This text discusses the proof that every third Fibonacci number is even, using mathematical induction. Let’s break it down step by step for clarity:

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Part a: Fibonacci sequence

The sequence $f_1, f_2, f_3, \dots$ is defined as:
$f_1 = 1, f_2 = 1, f_3 = 2, f_4 = 3, f_5 = 5, f_6 = 8, f_7 = 13, f_8 = 21, f_9 = 34, f_{10} = 55, f_{11} = 89, f_{12} = 144$

From this, we observe a pattern:

• Every third Fibonacci number (e.g., $f_3 = 2, f_6 = 8, f_9 = 34, f_{12} = 144$) is even, while the others are odd.

The task is to prove this property for all natural numbers $n$.

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Part b: Proof by induction

We aim to prove that $f_{3n}$ is even, and $f_{3n - 2}$ and $f_{3n - 1}$ are odd for any natural number $n$.

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1. Base Case (n = 1):

From the sequence:

• $f_3 = 2$ (even),
• $f_1 = 1$ (odd),
• $f_2 = 1$ (odd).

The statement holds for $n = 1$.

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2. Induction Hypothesis:

Assume that for some $k \geq 1$, the following is true:

• $f_{3k-2}$ is odd,
• $f_{3k-1}$ is odd,
• $f_{3k}$ is even.

We must show that this implies:

• $f_{3(k+1)-2}$ is odd,
• $f_{3(k+1)-1}$ is odd,
• $f_{3(k+1)}$ is even.

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3. Induction Step:

The Fibonacci recurrence relation states:
$f_{n+2} = f_{n+1} + f_n$

Let’s compute $f_{3(k+1)-2}, f_{3(k+1)-1}, f_{3(k+1)}$:

1. $f_{3(k+1)-2} = f_{3k+1} = f_{3k} + f_{3k-1}$:

   • $f_{3k}$ is even, $f_{3k-1}$ is odd, so their sum is odd.
   • Hence, $f_{3k+1}$ is odd.

2. $f_{3(k+1)-1} = f_{3k+2} = f_{3k+1} + f_{3k}$:

   • $f_{3k+1}$ is odd, $f_{3k}$ is even, so their sum is odd.
   • Hence, $f_{3k+2}$ is odd.

3. $f_{3(k+1)} = f_{3k+3} = f_{3k+2} + f_{3k+1}$:

   • Both $f_{3k+2}$ and $f_{3k+1}$ are odd, so their sum is even.
   • Hence, $f_{3k+3}$ is even.

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Conclusion:

By induction, $f_{3n}$ is even for all $n$, and $f_{3n-2}$, $f_{3n-1}$ are odd for all $n$.

This completes the proof.

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Summary

• The proof uses induction on the structure of the Fibonacci sequence.
• It relies on the recurrence relation and the properties of odd/even numbers under addition.

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Would you like further explanation or examples?

Here are 5 related questions to deepen understanding:

1. What are the properties of Fibonacci numbers regarding divisibility?
2. How does the modulo operation help identify patterns in sequences?
3. Can you generalize patterns in Fibonacci numbers for other moduli, like mod 3 or mod 5?
4. How is mathematical induction used in other proofs for Fibonacci sequences?
5. Can the proof be adapted for Lucas numbers (a related sequence)?

Tip: Always test small cases when dealing with induction problems to identify patterns or confirm hypotheses.