Antibonacci Sequence #2

Number Theory Level 3

Regarding the previous problem in this set , we know the first 4 backwards Fibonacci numbers. Using the method that you learned, keep going backwards. Besides being is reversed order, how does the forwards sequence ( $F_n$ ) contrast to the backwards sequence ( $F_{-n}$ )?

Note for clarity : $-2n+1$ means a negative number that is odd, and $-2n$ means a negative number that is even.

All of the odd numbers in the backwards sequence are negative.

F_{-2n+1}

is always positive and

F_{-2n}

is always negative. All of the even numbers in the backwards sequence are negative. They are the same sequences.

F_{-2n+1}

is always negative and

F_{-2n}

is always positive.

1 solution

Blan Morrison
Jan 24, 2018

Using the method previously used, we can find the numbers in the sequence between $F_{-7}$ and $F_7$ : $13,~-8,~5,~-3,~2,~-1,~1,~0,~1,~1,~2,~3,~5,~8,~13$

If we observe, we can see that every other number in the backwards sequence is negative. Upon closer observation, we can see that every $(-2n+1)^\text{th}$ number is negative. This pattern will clearly continue, so $F_{-2n+1}$ is always positive and $F_{-2n}$ is always negative.

Antibonacci Sequence #2

1 solution

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