To derive the reduction ratio $\backslash (\backslash$frac$\{$L$\_$n$\}\{$L$\_0\}=\backslash$frac$\{1\}\{$F$\_$n$\}\backslash )$ for the Fibonacci search method, we need to understand how the length of the array changes with each iteration based on Fibonacci numbers.

### Definitions

$-\backslash ($ L$\_0\backslash )$: Initial length of the array.

$-\backslash ($ L$\_$n $\backslash )$: Length of the array at the $\backslash ($ n $\backslash )-$th iteration.

$-\backslash ($ F$\_$n $\backslash )$: The $\backslash ($ n $\backslash )-$th Fibonacci number.

### Initial Setup

The Fibonacci sequence is defined as:

$\backslash [$ F$\_0=0,\backslash$quad F$\_1=1,\backslash$quad F$\_$n $=$ F$\_\{$n$-1\}+$ F$\_\{$n$-2\}\backslash$quad $\backslash$text$\{$for$\}\backslash$quad n $\backslash$geq $2\backslash ]$

We start with the smallest Fibonacci number $\backslash ($ F$\_$m $\backslash )$ that is greater than or equal to the length of the array $\backslash ($ L$\_0\backslash ).$

### Reduction Process

At each step, the array is divided into two parts using the Fibonacci numbers:

$1.**$First Iteration$**$:

$-$ Split the array using $\backslash ($ F$\_\{$m$-2\}\backslash )$ as the pivot.

$-$ Depending on whether the target element is less than or greater than the pivot element, the search range is reduced to either $\backslash ($ F$\_\{$m$-2\}\backslash )$ or $\backslash ($ F$\_\{$m$-1\}-$ F$\_\{$m$-2\}=$ F$\_\{$m$-3\}\backslash ).$

$2.**$Subsequent Iterations$**$:

$-$ The process repeats, reducing the search range using Fibonacci numbers.

### Reduction Ratio

Let's denote the length of the array at the $\backslash ($ n $\backslash )-$th iteration as $\backslash ($ L$\_$n $\backslash ).$ By the properties of the Fibonacci search, each reduction step divides the array according to the Fibonacci numbers.

After $\backslash ($ n $\backslash )$ iterations, the array length $\backslash ($ L$\_$n $\backslash )$ can be expressed in terms of $\backslash ($ F$\_\{$m$-$n$\}\backslash ),$ where $\backslash ($ m $\backslash )$ is the original Fibonacci number chosen based on the initial array length.

We know that:

$\backslash [$ L$\_$n $=$ F$\_\{$m$-$n$\}\backslash ]$

$\backslash [$ L$\_0=$ F$\_$m $\backslash ]$

The reduction ratio after $\backslash ($ n $\backslash )$ iterations is:

$\backslash [\backslash$frac$\{$L$\_$n$\}\{$L$\_0\}=\backslash$frac$\{$F$\_\{$m$-$n$\}\}\{$F$\_$m$\}\backslash ]$

### Special Case: Final Iteration

In the final iteration, we consider the case when $\backslash ($ n $=$ m $\backslash )$:

$\backslash [$ L$\_$m $=$ F$\_0=1\backslash ]$

$\backslash [$ L$\_0=$ F$\_$m $\backslash ]$

Thus, the reduction ratio after $\backslash ($ m $\backslash )$ iterations is:

$\backslash [\backslash$frac$\{$L$\_$m$\}\{$L$\_0\}=\backslash$frac$\{1\}\{$F$\_$m$\}\backslash ]$

### General Case

For the general case, after $\backslash ($ n $\backslash )$ iterations:

$\backslash [\backslash$frac$\{$L$\_$n$\}\{$L$\_0\}=\backslash$frac$\{$F$\_\{$m$-$n$\}\}\{$F$\_$m$\}\backslash ]$

Given that $\backslash ($ F$\_\{$m$-$n$\}\backslash$approx $1\backslash )$ for large $\backslash ($ n $\backslash )$ and assuming $\backslash ($ n $\backslash )$ approaches $\backslash ($ m $\backslash ),$ we get the reduction ratio as:

$\backslash [\backslash$frac$\{$L$\_$n$\}\{$L$\_0\}\backslash$approx $\backslash$frac$\{1\}\{$F$\_$n$\}\backslash ]$

This shows how the length of the array decreases with each iteration, and specifically, in the final iteration, the array length is reduced by a factor of the corresponding Fibonacci number.