Fibonacci Addition Identity for Fibonacci Numbers Separated by 3 Terms The Fibonacci Addition Ident

Salvador Bush 2022-07-04 Answered
Fibonacci Addition Identity for Fibonacci Numbers Separated by 3 Terms
The Fibonacci Addition Identity states that: F n = F m F n m + 1 + F m 1 F n m . This was useful in showing that: F i + k = F k 2 F i + 1 + F k 1 F i + 2 . However, I would like to use this result to express the same for F i and F i + 3 , where we can express F i + k in some linear combination of F i and F i + 3 . Is there any way to do this? I haven't been able to make use of the typical Fibonacci substitutions to make any progress.
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Answers (1)

Sariah Glover
Answered 2022-07-05 Author has 16 answers
Step 1
We have F i = F i + 2 F i + 1 and F i + 3 = F i + 1 + F i + 2 ; we can solve these two equations to get F i + 1 , F i + 2 in terms of F i , F i + 3 instead. This gives us F i + 2 = 1 2 ( F i + F i + 3 ) and F i + 1 = 1 2 ( F i + 3 F i ).
Step 2
Now substitute this into the identity you've already found:
F i + k = F k 2 F i + 1 + F k 1 F i + 2 = F k 2 ( F i + 3 F i 2 ) + F k 1 ( F i + F i + 3 2 ) = ( F k 1 F k 2 2 ) F i + ( F k 2 + F k 1 2 ) F i + 3 = 1 2 F k 3 F i + 1 2 F k F i + 3 .

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