 # 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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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 ${F}_{i+2}=\frac{1}{2}\left({F}_{i}+{F}_{i+3}\right)$ and ${F}_{i+1}=\frac{1}{2}\left({F}_{i+3}-{F}_{i}\right)$.
Step 2
Now substitute this into the identity you've already found:
$\begin{array}{rl}{F}_{i+k}& ={F}_{k-2}{F}_{i+1}+{F}_{k-1}{F}_{i+2}\\ & ={F}_{k-2}\left(\frac{{F}_{i+3}-{F}_{i}}{2}\right)+{F}_{k-1}\left(\frac{{F}_{i}+{F}_{i+3}}{2}\right)\\ & =\left(\frac{{F}_{k-1}-{F}_{k-2}}{2}\right){F}_{i}+\left(\frac{{F}_{k-2}+{F}_{k-1}}{2}\right){F}_{i+3}\\ & =\frac{1}{2}{F}_{k-3}{F}_{i}+\frac{1}{2}{F}_{k}{F}_{i+3}.\end{array}$

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