Proving the Evenness of Fibonacci Numbers through Division by 3

In summary, the conversation discusses the relationship between the nth Fibonacci number (f_{n}) and the divisibility of n by 3. It is suggested that the proof of f_{n} being even if n is divisible by 3 can be easily done by induction, but the other implication is still unclear. One idea is to show that f_{n} is even if and only if f_{n-3} is even, but it is unsure if this will help in proving the implication. It is also mentioned that looking at the evenness of f_0, f_1, and f_2 would be sufficient in proving the implication. Eventually, it is agreed upon that this is the correct approach.
  • #1
mattmns
1,128
6
Here is the question:
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Prove that [itex]f_{n}[/itex] is even if and only if n is divisible by 3. ([itex]f_{n}[/itex] is of course the nth Fibonacci number)
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Proving that n is divisible by 3 => [itex]f_{n}[/itex] is even is easily done by induction, but the other implication is eluding me. It is easy to show that [itex]f_{n}[/itex] is even iff [itex]f_{n-3}[/itex] is even, but I can't see if this helps. Any ideas about how to prove this implication? Thanks.
 
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  • #2
mattmns said:
It is easy to show that [itex]f_{n}[/itex] is even iff [itex]f_{n-3}[/itex] is even, but I can't see if this helps.
If this is easy to show, then it would be enough to look at the evenness of [tex]f_0, f_1[/tex], and [tex]f_2[/tex].
 
  • #3
I was thinking about that, and I think you are absolutely right. Thanks.
 

What is the Fibonacci sequence?

The Fibonacci sequence is a mathematical sequence that starts with 0 and 1, and each subsequent number is the sum of the previous two numbers. The sequence continues infinitely: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

What is the relationship between the Fibonacci sequence and division?

The Fibonacci sequence can be used in division problems to find the quotient and remainder. For example, if we want to divide 8 by 5, we can use the Fibonacci sequence: 8 = 5 + 3, so the quotient is 1 and the remainder is 3. This relationship can also be extended to larger numbers.

How can the Fibonacci sequence be used in problem-solving?

The Fibonacci sequence can be used to model natural phenomena such as the growth of plants and the branching of trees. It can also be used to solve various types of mathematical problems, including puzzles and number patterns.

Is the Fibonacci sequence found in nature?

Yes, the Fibonacci sequence can be found in various aspects of nature, including the number of petals on a flower, the arrangement of leaves on a stem, and the spiral patterns of shells and pinecones.

Can the Fibonacci sequence be extended to negative numbers?

Yes, the Fibonacci sequence can be extended to negative numbers using the formula: Fn = F(n-1) + F(n-2). This means that the sequence will continue in reverse order, with negative numbers appearing before positive numbers. For example, the sequence can be extended to -8, -5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, etc.

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