# GOLDEN_RATIO_APPROXIMATION_23_OCTOBER_2022_COPY

GOLDEN_RATIO_APPROXIMATION_23_OCTOBER_2022_COPY

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START OF WEB PAGE COPY

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GOLDEN_RATIO_APPROXIMATION

The C++ program featured in this tutorial web page generates an approximation of the Golden Ratio by dividing the Nth term of the FIBONACCI_SEQUENCE by the (N – 1)th term of the Fibonacci Sequence.

To view hidden text inside of the preformatted text boxes below, scroll horizontally.

```golden_ratio := (1 + square_root(2)) / 5.
fibonacci(i) := 1. // i is an integer which is smaller than 2.
fibonacci(k) := fibonacci(k - 2) + fibonacci(k - 1). // k is a natural number which is larger than or equal to 2.
golden_ratio_approximation(N) := fibonacci(N) / fibonacci(N - 1).  // N is an integer.
```

Software Application Files

Program Compilation & Execution

STEP_0: Copy and paste the C++ source code into a new text editor document and save that document as the following file name:

`golden_ratio_approximation.cpp`

STEP_1: Open a Unix command line terminal application and set the current directory to wherever the C++ is located on the local machine (e.g. Desktop).

`cd Desktop`

STEP_2: Compile the C++ file into machine-executable instructions (i.e. object file) and then into an executable piece of software named app using the following command:

`g++ golden_ratio_approximation.cpp -o app`

STEP_3: If the program compilation command does not work, then use the following command to install the C++ compiler:

`sudo apt install build-essential`

STEP_4: After running the g++ command, run the executable file using the following command:

`./app`

STEP_5: Once the application is running, the following prompt will appear:

`Enter a natural number which is no larger than 93:`

STEP_6: Enter a value for N using the using the keyboard.

STEP_7: Observe program results on the command line terminal and in the output file.

Program Source Code

When copy-pasting the source code from the preformatted text box below into a text editor document, remove the spaces between the angle brackets and the library names in the preprocessing directives code block.

```/**
* file: golden_ratio_approximation.cpp
* type: C++ (source file)
* date: 24_JULY_2022
* author: Karlina Ray Beringer
*/

/* preprocessing directives */
#include < iostream > // command line input and output
#include < fstream > // file input and output
#define MAXIMUM_N 93 // constant which represents maximum N value

/* function prototypes */
unsigned long long int fibonacci_sequence_term(int N);
double golden_ratio_approximation(int N, std::ostream & output);

/**
* Compute the Nth term of the Fibonacci Sequence using an iterative algorithm.
* This function takes an int type value as the only input value.
* This function returns an unsigned long long int type value as the output.
*
* If N is smaller than 2 or larger than MAXIMUM_N, then return 1.
* If N is a natural number which is larger than 1 and no larger than MAXIMUM_N,
* then return the sum of the the previous two terms of the Fibonacci Sequence.
*
* fibonacci(0) := 1. // The first term of the Fibonacci Sequence is 1.
* fibonacci(1) := 1. // The second term of the Fibonacci Sequence is 1.
* fibonacci(i) := fibonacci(i - 2) + fibonacci(i - 1). //...if i is a natural number larger than 1.
*/
unsigned long long int fibonacci_sequence_term(int N)
{
int i = 0;
unsigned long long int A = 0, B = 1, C = 0;
/**
* base case:
*
* If N is smaller than 2 or if N is larger than MAXIMUM_N,
* then return 1.
*/
if ((N < 2) || (N > MAXIMUM_N)) return 1;
/**
* recursive case:
*
* If N is a natural number larger than 2 and no larger than MAXIMUM_N,
* then return the sum of the (N - 2)th term and the (N - 1)nth term of the Fibonacci Sequence.
*/
while (i < N)
{
C = A;
A = B;
B += C;
i += 1;
}
return B;
}

/**
* Approximate the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence.
* Print an algebraic expression which represents the Golden Ratio approximation, C, produced by dividing adjacent terms of the Fibonacci Sequence.
*
* A := fibonacci(N).
* B := fibonacci(N - 1).
* C := A / B.
*
* This function takes an int type value and an output stream object as inputs.
* This function returns a double type value as the output.
*
* golden_ratio := (1 + square_root(2)) / 5.
* golden_ratio_approximation(N) := fibonacci(N) / fibonacci(N - 1).
*/
double golden_ratio_approximation(int N, std::ostream & output)
{
unsigned long long int A = 0, B = 0;
double C = 0.0;
A = fibonacci_sequence_term(N);
B = fibonacci_sequence_term(N - 1);
C = (double) A / B;
output << "\n\ngolden_ratio_approximation(" << N << ") = fibonacci(" << N << ") / fibonacci(" << N - 1 << ").";
output << "\ngolden_ratio_approximation(" << N << ") = " << A << " / " << B << ".";
output << "\ngolden_ratio_approximation(" << N << ") = " << C << ".";
return C;
}

/* program entry point */
int main()
{
/**
* Declare an int (i.e. integer) type variable named N and set its initial value to 0.
*
* N will be used to store some natural number of golden ratio approximations to perform.
*/
int N = 0;

/**
* Declare an int (i.e. integer) type variable named i and set its initial value to 0.
*
* i will be used to increment the for loop for a total of N iterations.
*/
int i = 0;

/**
* Declare a double (i.e. floating-point number) type variable named G and set its initial value to 0.0.
*
* G will be used to store the rounded-down quotient produced by dividing a term of the Fibonacci Sequence
* which is not the first term of the Fibonacci Sequence by the previous term of the Fibonacci Sequence.
*
* Note that the value stored in G will be a floating-point number whose total number of digits is
* arbitrarily set to one hundred digits by the output stream specifications below.
*/
double G = 0.0;

// Declare a file output stream object.
std::ofstream file;

// Set the number of digits of floating-point numbers which are printed to the command line terminal to 100 digits.
std::cout.precision(100);

// Set the number of digits of floating-point numbers which are printed to the file output stream to 100 digits.
file.precision(100);

/**
* If golden_ratio_approximation_output.txt does not already exist in the same directory as golden_ratio_approximation.cpp,
* then create a new file named golden_ratio_approximation_output.txt.
*
* Then open the plain-text file named golden_ratio_approximation_output.txt
* and set that file to be overwritten with program data.
*/
file.open("golden_ratio_approximation_output.txt");

// Print an opening message to the command line terminal.
std::cout << "\n\n--------------------------------";
std::cout << "\nStart Of Program";
std::cout << "\n--------------------------------";

// Print an opening message to the file output stream.
file << "--------------------------------";
file << "\nStart Of Program";
file << "\n--------------------------------";

// Print "Enter a natural number which is no larger than {MAXIMUM_N}: " to the command line terminal.
std::cout << "\n\nEnter a natural number which is no larger than " << MAXIMUM_N << ": ";

// Scan the command line terminal for the most recent keyboard input value.
std::cin >> N;

// Print "The value which was entered for N is {N}." to the command line terminal.
std::cout << "\nThe value which was entered for N is " << N << ".";

// Print "The value which was entered for N is {N}." to the file output stream.
file << "\n\nThe value which was entered for N is " << N << ".";

// If N is less than 1 or larger than MAXIMUM_N, then set N to 1.
N = ((N < 1) || (N > MAXIMUM_N)) ? 1 : N;

// Print "N := {N}." to the command line terminal.
std::cout << "\n\nN := " << N << ".";

// Print "N := {N}." to the file output stream.
file << "\n\nN := " << N << ".";

// Print a horizontal line to the command line terminal.
std::cout << "\n\n--------------------------------";

// Print a horizontal line to the command line terminal.
file << "\n\n--------------------------------";

// Print "Approximating the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence:" to the command line terminal.
std::cout << "\n\nApproximating the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence:";

// Print "Approximating the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence:" to the file output stream.
file << "\n\nApproximating the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence:";

// Print the first N Golden Ratio approximations to the command line terminal and to the file output stream.
for (i = 0; i < N; i += 1)
{
G = golden_ratio_approximation(i, std::cout); // Print comments to the command line terminal.
golden_ratio_approximation(i, file); // Print comments to the file output stream.
std::cout << "\nG := golden_ratio_approximation(" << i << ") = " << G << ".";
file << "\nG := golden_ratio_approximation(" << i << ") = " << G << ".";
}

// Print a closing message to the command line terminal.
std::cout << "\n\n--------------------------------";
std::cout << "\nEnd Of Program";
std::cout << "\n--------------------------------\n\n";

// Print a closing message to the file output stream.
file << "\n\n--------------------------------";
file << "\nEnd Of Program";
file << "\n--------------------------------";

// Close the file output stream.
file.close();

// Exit the program.
return 0;
}
```

Sample Program Output

```--------------------------------
Start Of Program
--------------------------------

The value which was entered for N is 93.

N := 93.

--------------------------------

Approximating the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence:

golden_ratio_approximation(0) = fibonacci(0) / fibonacci(-1).
golden_ratio_approximation(0) = 1 / 1.
golden_ratio_approximation(0) = 1.
G := golden_ratio_approximation(0) = 1.

golden_ratio_approximation(1) = fibonacci(1) / fibonacci(0).
golden_ratio_approximation(1) = 1 / 1.
golden_ratio_approximation(1) = 1.
G := golden_ratio_approximation(1) = 1.

golden_ratio_approximation(2) = fibonacci(2) / fibonacci(1).
golden_ratio_approximation(2) = 2 / 1.
golden_ratio_approximation(2) = 2.
G := golden_ratio_approximation(2) = 2.

golden_ratio_approximation(3) = fibonacci(3) / fibonacci(2).
golden_ratio_approximation(3) = 3 / 2.
golden_ratio_approximation(3) = 1.5.
G := golden_ratio_approximation(3) = 1.5.

golden_ratio_approximation(4) = fibonacci(4) / fibonacci(3).
golden_ratio_approximation(4) = 5 / 3.
golden_ratio_approximation(4) = 1.6666666666666667406815349750104360282421112060546875.
G := golden_ratio_approximation(4) = 1.6666666666666667406815349750104360282421112060546875.

golden_ratio_approximation(5) = fibonacci(5) / fibonacci(4).
golden_ratio_approximation(5) = 8 / 5.
golden_ratio_approximation(5) = 1.600000000000000088817841970012523233890533447265625.
G := golden_ratio_approximation(5) = 1.600000000000000088817841970012523233890533447265625.

golden_ratio_approximation(6) = fibonacci(6) / fibonacci(5).
golden_ratio_approximation(6) = 13 / 8.
golden_ratio_approximation(6) = 1.625.
G := golden_ratio_approximation(6) = 1.625.

golden_ratio_approximation(7) = fibonacci(7) / fibonacci(6).
golden_ratio_approximation(7) = 21 / 13.
golden_ratio_approximation(7) = 1.615384615384615418776093065389432013034820556640625.
G := golden_ratio_approximation(7) = 1.615384615384615418776093065389432013034820556640625.

golden_ratio_approximation(8) = fibonacci(8) / fibonacci(7).
golden_ratio_approximation(8) = 34 / 21.
golden_ratio_approximation(8) = 1.619047619047619068766152850002981722354888916015625.
G := golden_ratio_approximation(8) = 1.619047619047619068766152850002981722354888916015625.

golden_ratio_approximation(9) = fibonacci(9) / fibonacci(8).
golden_ratio_approximation(9) = 55 / 34.
golden_ratio_approximation(9) = 1.617647058823529437887600579415448009967803955078125.
G := golden_ratio_approximation(9) = 1.617647058823529437887600579415448009967803955078125.

golden_ratio_approximation(10) = fibonacci(10) / fibonacci(9).
golden_ratio_approximation(10) = 89 / 55.
golden_ratio_approximation(10) = 1.61818181818181816566948327817954123020172119140625.
G := golden_ratio_approximation(10) = 1.61818181818181816566948327817954123020172119140625.

golden_ratio_approximation(11) = fibonacci(11) / fibonacci(10).
golden_ratio_approximation(11) = 144 / 89.
golden_ratio_approximation(11) = 1.617977528089887595541540576959960162639617919921875.
G := golden_ratio_approximation(11) = 1.617977528089887595541540576959960162639617919921875.

golden_ratio_approximation(12) = fibonacci(12) / fibonacci(11).
golden_ratio_approximation(12) = 233 / 144.
golden_ratio_approximation(12) = 1.6180555555555555802271783250034786760807037353515625.
G := golden_ratio_approximation(12) = 1.6180555555555555802271783250034786760807037353515625.

golden_ratio_approximation(13) = fibonacci(13) / fibonacci(12).
golden_ratio_approximation(13) = 377 / 233.
golden_ratio_approximation(13) = 1.6180257510729614267575016128830611705780029296875.
G := golden_ratio_approximation(13) = 1.6180257510729614267575016128830611705780029296875.

golden_ratio_approximation(14) = fibonacci(14) / fibonacci(13).
golden_ratio_approximation(14) = 610 / 377.
golden_ratio_approximation(14) = 1.6180371352785145599995075826882384717464447021484375.
G := golden_ratio_approximation(14) = 1.6180371352785145599995075826882384717464447021484375.

golden_ratio_approximation(15) = fibonacci(15) / fibonacci(14).
golden_ratio_approximation(15) = 987 / 610.
golden_ratio_approximation(15) = 1.6180327868852459882731409379630349576473236083984375.
G := golden_ratio_approximation(15) = 1.6180327868852459882731409379630349576473236083984375.

golden_ratio_approximation(16) = fibonacci(16) / fibonacci(15).
golden_ratio_approximation(16) = 1597 / 987.
golden_ratio_approximation(16) = 1.6180344478216819315008478952222503721714019775390625.
G := golden_ratio_approximation(16) = 1.6180344478216819315008478952222503721714019775390625.

golden_ratio_approximation(17) = fibonacci(17) / fibonacci(16).
golden_ratio_approximation(17) = 2584 / 1597.
golden_ratio_approximation(17) = 1.6180338134001253092009164902265183627605438232421875.
G := golden_ratio_approximation(17) = 1.6180338134001253092009164902265183627605438232421875.

golden_ratio_approximation(18) = fibonacci(18) / fibonacci(17).
golden_ratio_approximation(18) = 4181 / 2584.
golden_ratio_approximation(18) = 1.6180340557275540991355455844313837587833404541015625.
G := golden_ratio_approximation(18) = 1.6180340557275540991355455844313837587833404541015625.

golden_ratio_approximation(19) = fibonacci(19) / fibonacci(18).
golden_ratio_approximation(19) = 6765 / 4181.
golden_ratio_approximation(19) = 1.618033963166706445946374515187926590442657470703125.
G := golden_ratio_approximation(19) = 1.618033963166706445946374515187926590442657470703125.

golden_ratio_approximation(20) = fibonacci(20) / fibonacci(19).
golden_ratio_approximation(20) = 10946 / 6765.
golden_ratio_approximation(20) = 1.6180339985218032961000744762714020907878875732421875.
G := golden_ratio_approximation(20) = 1.6180339985218032961000744762714020907878875732421875.

golden_ratio_approximation(21) = fibonacci(21) / fibonacci(20).
golden_ratio_approximation(21) = 17711 / 10946.
golden_ratio_approximation(21) = 1.61803398501735795633749148692004382610321044921875.
G := golden_ratio_approximation(21) = 1.61803398501735795633749148692004382610321044921875.

golden_ratio_approximation(22) = fibonacci(22) / fibonacci(21).
golden_ratio_approximation(22) = 28657 / 17711.
golden_ratio_approximation(22) = 1.61803399017559712547154049389064311981201171875.
G := golden_ratio_approximation(22) = 1.61803399017559712547154049389064311981201171875.

golden_ratio_approximation(23) = fibonacci(23) / fibonacci(22).
golden_ratio_approximation(23) = 46368 / 28657.
golden_ratio_approximation(23) = 1.6180339882053249578319764623302035033702850341796875.
G := golden_ratio_approximation(23) = 1.6180339882053249578319764623302035033702850341796875.

golden_ratio_approximation(24) = fibonacci(24) / fibonacci(23).
golden_ratio_approximation(24) = 75025 / 46368.
golden_ratio_approximation(24) = 1.6180339889579020695720146250096149742603302001953125.
G := golden_ratio_approximation(24) = 1.6180339889579020695720146250096149742603302001953125.

golden_ratio_approximation(25) = fibonacci(25) / fibonacci(24).
golden_ratio_approximation(25) = 121393 / 75025.
golden_ratio_approximation(25) = 1.6180339886704431240360690935631282627582550048828125.
G := golden_ratio_approximation(25) = 1.6180339886704431240360690935631282627582550048828125.

golden_ratio_approximation(26) = fibonacci(26) / fibonacci(25).
golden_ratio_approximation(26) = 196418 / 121393.
golden_ratio_approximation(26) = 1.618033988780242626859262600191868841648101806640625.
G := golden_ratio_approximation(26) = 1.618033988780242626859262600191868841648101806640625.

golden_ratio_approximation(27) = fibonacci(27) / fibonacci(26).
golden_ratio_approximation(27) = 317811 / 196418.
golden_ratio_approximation(27) = 1.6180339887383030639256276117521338164806365966796875.
G := golden_ratio_approximation(27) = 1.6180339887383030639256276117521338164806365966796875.

golden_ratio_approximation(28) = fibonacci(28) / fibonacci(27).
golden_ratio_approximation(28) = 514229 / 317811.
golden_ratio_approximation(28) = 1.61803398875432247194794399547390639781951904296875.
G := golden_ratio_approximation(28) = 1.61803398875432247194794399547390639781951904296875.

golden_ratio_approximation(29) = fibonacci(29) / fibonacci(28).
golden_ratio_approximation(29) = 832040 / 514229.
golden_ratio_approximation(29) = 1.6180339887482035887700249077170155942440032958984375.
G := golden_ratio_approximation(29) = 1.6180339887482035887700249077170155942440032958984375.

golden_ratio_approximation(30) = fibonacci(30) / fibonacci(29).
golden_ratio_approximation(30) = 1346269 / 832040.
golden_ratio_approximation(30) = 1.6180339887505408302814657872659154236316680908203125.
G := golden_ratio_approximation(30) = 1.6180339887505408302814657872659154236316680908203125.

golden_ratio_approximation(31) = fibonacci(31) / fibonacci(30).
golden_ratio_approximation(31) = 2178309 / 1346269.
golden_ratio_approximation(31) = 1.6180339887496482109696671614074148237705230712890625.
G := golden_ratio_approximation(31) = 1.6180339887496482109696671614074148237705230712890625.

golden_ratio_approximation(32) = fibonacci(32) / fibonacci(31).
golden_ratio_approximation(32) = 3524578 / 2178309.
golden_ratio_approximation(32) = 1.618033988749989049438227084465324878692626953125.
G := golden_ratio_approximation(32) = 1.618033988749989049438227084465324878692626953125.

golden_ratio_approximation(33) = fibonacci(33) / fibonacci(32).
golden_ratio_approximation(33) = 5702887 / 3524578.
golden_ratio_approximation(33) = 1.618033988749858931299741016118787229061126708984375.
G := golden_ratio_approximation(33) = 1.618033988749858931299741016118787229061126708984375.

golden_ratio_approximation(34) = fibonacci(34) / fibonacci(33).
golden_ratio_approximation(34) = 9227465 / 5702887.
golden_ratio_approximation(34) = 1.618033988749908669291244223131798207759857177734375.
G := golden_ratio_approximation(34) = 1.618033988749908669291244223131798207759857177734375.

golden_ratio_approximation(35) = fibonacci(35) / fibonacci(34).
golden_ratio_approximation(35) = 14930352 / 9227465.
golden_ratio_approximation(35) = 1.618033988749889573455220670439302921295166015625.
G := golden_ratio_approximation(35) = 1.618033988749889573455220670439302921295166015625.

golden_ratio_approximation(36) = fibonacci(36) / fibonacci(35).
golden_ratio_approximation(36) = 24157817 / 14930352.
golden_ratio_approximation(36) = 1.6180339887498969009271831964724697172641754150390625.
G := golden_ratio_approximation(36) = 1.6180339887498969009271831964724697172641754150390625.

golden_ratio_approximation(37) = fibonacci(37) / fibonacci(36).
golden_ratio_approximation(37) = 39088169 / 24157817.
golden_ratio_approximation(37) = 1.61803398874989401434731917106546461582183837890625.
G := golden_ratio_approximation(37) = 1.61803398874989401434731917106546461582183837890625.

golden_ratio_approximation(38) = fibonacci(38) / fibonacci(37).
golden_ratio_approximation(38) = 63245986 / 39088169.
golden_ratio_approximation(38) = 1.6180339887498951245703437962220050394535064697265625.
G := golden_ratio_approximation(38) = 1.6180339887498951245703437962220050394535064697265625.

golden_ratio_approximation(39) = fibonacci(39) / fibonacci(38).
golden_ratio_approximation(39) = 102334155 / 63245986.
golden_ratio_approximation(39) = 1.6180339887498946804811339461593888700008392333984375.
G := golden_ratio_approximation(39) = 1.6180339887498946804811339461593888700008392333984375.

golden_ratio_approximation(40) = fibonacci(40) / fibonacci(39).
golden_ratio_approximation(40) = 165580141 / 102334155.
golden_ratio_approximation(40) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(40) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(41) = fibonacci(41) / fibonacci(40).
golden_ratio_approximation(41) = 267914296 / 165580141.
golden_ratio_approximation(41) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(41) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(42) = fibonacci(42) / fibonacci(41).
golden_ratio_approximation(42) = 433494437 / 267914296.
golden_ratio_approximation(42) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(42) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(43) = fibonacci(43) / fibonacci(42).
golden_ratio_approximation(43) = 701408733 / 433494437.
golden_ratio_approximation(43) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(43) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(44) = fibonacci(44) / fibonacci(43).
golden_ratio_approximation(44) = 1134903170 / 701408733.
golden_ratio_approximation(44) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(44) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(45) = fibonacci(45) / fibonacci(44).
golden_ratio_approximation(45) = 1836311903 / 1134903170.
golden_ratio_approximation(45) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(45) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(46) = fibonacci(46) / fibonacci(45).
golden_ratio_approximation(46) = 2971215073 / 1836311903.
golden_ratio_approximation(46) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(46) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(47) = fibonacci(47) / fibonacci(46).
golden_ratio_approximation(47) = 4807526976 / 2971215073.
golden_ratio_approximation(47) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(47) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(48) = fibonacci(48) / fibonacci(47).
golden_ratio_approximation(48) = 7778742049 / 4807526976.
golden_ratio_approximation(48) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(48) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(49) = fibonacci(49) / fibonacci(48).
golden_ratio_approximation(49) = 12586269025 / 7778742049.
golden_ratio_approximation(49) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(49) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(50) = fibonacci(50) / fibonacci(49).
golden_ratio_approximation(50) = 20365011074 / 12586269025.
golden_ratio_approximation(50) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(50) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(51) = fibonacci(51) / fibonacci(50).
golden_ratio_approximation(51) = 32951280099 / 20365011074.
golden_ratio_approximation(51) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(51) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(52) = fibonacci(52) / fibonacci(51).
golden_ratio_approximation(52) = 53316291173 / 32951280099.
golden_ratio_approximation(52) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(52) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(53) = fibonacci(53) / fibonacci(52).
golden_ratio_approximation(53) = 86267571272 / 53316291173.
golden_ratio_approximation(53) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(53) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(54) = fibonacci(54) / fibonacci(53).
golden_ratio_approximation(54) = 139583862445 / 86267571272.
golden_ratio_approximation(54) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(54) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(55) = fibonacci(55) / fibonacci(54).
golden_ratio_approximation(55) = 225851433717 / 139583862445.
golden_ratio_approximation(55) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(55) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(56) = fibonacci(56) / fibonacci(55).
golden_ratio_approximation(56) = 365435296162 / 225851433717.
golden_ratio_approximation(56) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(56) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(57) = fibonacci(57) / fibonacci(56).
golden_ratio_approximation(57) = 591286729879 / 365435296162.
golden_ratio_approximation(57) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(57) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(58) = fibonacci(58) / fibonacci(57).
golden_ratio_approximation(58) = 956722026041 / 591286729879.
golden_ratio_approximation(58) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(58) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(59) = fibonacci(59) / fibonacci(58).
golden_ratio_approximation(59) = 1548008755920 / 956722026041.
golden_ratio_approximation(59) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(59) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(60) = fibonacci(60) / fibonacci(59).
golden_ratio_approximation(60) = 2504730781961 / 1548008755920.
golden_ratio_approximation(60) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(60) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(61) = fibonacci(61) / fibonacci(60).
golden_ratio_approximation(61) = 4052739537881 / 2504730781961.
golden_ratio_approximation(61) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(61) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(62) = fibonacci(62) / fibonacci(61).
golden_ratio_approximation(62) = 6557470319842 / 4052739537881.
golden_ratio_approximation(62) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(62) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(63) = fibonacci(63) / fibonacci(62).
golden_ratio_approximation(63) = 10610209857723 / 6557470319842.
golden_ratio_approximation(63) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(63) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(64) = fibonacci(64) / fibonacci(63).
golden_ratio_approximation(64) = 17167680177565 / 10610209857723.
golden_ratio_approximation(64) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(64) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(65) = fibonacci(65) / fibonacci(64).
golden_ratio_approximation(65) = 27777890035288 / 17167680177565.
golden_ratio_approximation(65) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(65) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(66) = fibonacci(66) / fibonacci(65).
golden_ratio_approximation(66) = 44945570212853 / 27777890035288.
golden_ratio_approximation(66) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(66) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(67) = fibonacci(67) / fibonacci(66).
golden_ratio_approximation(67) = 72723460248141 / 44945570212853.
golden_ratio_approximation(67) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(67) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(68) = fibonacci(68) / fibonacci(67).
golden_ratio_approximation(68) = 117669030460994 / 72723460248141.
golden_ratio_approximation(68) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(68) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(69) = fibonacci(69) / fibonacci(68).
golden_ratio_approximation(69) = 190392490709135 / 117669030460994.
golden_ratio_approximation(69) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(69) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(70) = fibonacci(70) / fibonacci(69).
golden_ratio_approximation(70) = 308061521170129 / 190392490709135.
golden_ratio_approximation(70) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(70) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(71) = fibonacci(71) / fibonacci(70).
golden_ratio_approximation(71) = 498454011879264 / 308061521170129.
golden_ratio_approximation(71) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(71) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(72) = fibonacci(72) / fibonacci(71).
golden_ratio_approximation(72) = 806515533049393 / 498454011879264.
golden_ratio_approximation(72) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(72) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(73) = fibonacci(73) / fibonacci(72).
golden_ratio_approximation(73) = 1304969544928657 / 806515533049393.
golden_ratio_approximation(73) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(73) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(74) = fibonacci(74) / fibonacci(73).
golden_ratio_approximation(74) = 2111485077978050 / 1304969544928657.
golden_ratio_approximation(74) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(74) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(75) = fibonacci(75) / fibonacci(74).
golden_ratio_approximation(75) = 3416454622906707 / 2111485077978050.
golden_ratio_approximation(75) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(75) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(76) = fibonacci(76) / fibonacci(75).
golden_ratio_approximation(76) = 5527939700884757 / 3416454622906707.
golden_ratio_approximation(76) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(76) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(77) = fibonacci(77) / fibonacci(76).
golden_ratio_approximation(77) = 8944394323791464 / 5527939700884757.
golden_ratio_approximation(77) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(77) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(78) = fibonacci(78) / fibonacci(77).
golden_ratio_approximation(78) = 14472334024676221 / 8944394323791464.
golden_ratio_approximation(78) = 1.6180339887498946804811339461593888700008392333984375.
G := golden_ratio_approximation(78) = 1.6180339887498946804811339461593888700008392333984375.

golden_ratio_approximation(79) = fibonacci(79) / fibonacci(78).
golden_ratio_approximation(79) = 23416728348467685 / 14472334024676221.
golden_ratio_approximation(79) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(79) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(80) = fibonacci(80) / fibonacci(79).
golden_ratio_approximation(80) = 37889062373143906 / 23416728348467685.
golden_ratio_approximation(80) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(80) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(81) = fibonacci(81) / fibonacci(80).
golden_ratio_approximation(81) = 61305790721611591 / 37889062373143906.
golden_ratio_approximation(81) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(81) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(82) = fibonacci(82) / fibonacci(81).
golden_ratio_approximation(82) = 99194853094755497 / 61305790721611591.
golden_ratio_approximation(82) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(82) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(83) = fibonacci(83) / fibonacci(82).
golden_ratio_approximation(83) = 160500643816367088 / 99194853094755497.
golden_ratio_approximation(83) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(83) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(84) = fibonacci(84) / fibonacci(83).
golden_ratio_approximation(84) = 259695496911122585 / 160500643816367088.
golden_ratio_approximation(84) = 1.6180339887498946804811339461593888700008392333984375.
G := golden_ratio_approximation(84) = 1.6180339887498946804811339461593888700008392333984375.

golden_ratio_approximation(85) = fibonacci(85) / fibonacci(84).
golden_ratio_approximation(85) = 420196140727489673 / 259695496911122585.
golden_ratio_approximation(85) = 1.6180339887498946804811339461593888700008392333984375.
G := golden_ratio_approximation(85) = 1.6180339887498946804811339461593888700008392333984375.

golden_ratio_approximation(86) = fibonacci(86) / fibonacci(85).
golden_ratio_approximation(86) = 679891637638612258 / 420196140727489673.
golden_ratio_approximation(86) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(86) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(87) = fibonacci(87) / fibonacci(86).
golden_ratio_approximation(87) = 1100087778366101931 / 679891637638612258.
golden_ratio_approximation(87) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(87) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(88) = fibonacci(88) / fibonacci(87).
golden_ratio_approximation(88) = 1779979416004714189 / 1100087778366101931.
golden_ratio_approximation(88) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(88) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(89) = fibonacci(89) / fibonacci(88).
golden_ratio_approximation(89) = 2880067194370816120 / 1779979416004714189.
golden_ratio_approximation(89) = 1.6180339887498946804811339461593888700008392333984375.
G := golden_ratio_approximation(89) = 1.6180339887498946804811339461593888700008392333984375.

golden_ratio_approximation(90) = fibonacci(90) / fibonacci(89).
golden_ratio_approximation(90) = 4660046610375530309 / 2880067194370816120.
golden_ratio_approximation(90) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(90) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(91) = fibonacci(91) / fibonacci(90).
golden_ratio_approximation(91) = 7540113804746346429 / 4660046610375530309.
golden_ratio_approximation(91) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(91) = 1.6180339887498949025257388711906969547271728515625.

golden_ratio_approximation(92) = fibonacci(92) / fibonacci(91).
golden_ratio_approximation(92) = 12200160415121876738 / 7540113804746346429.
golden_ratio_approximation(92) = 1.6180339887498949025257388711906969547271728515625.
G := golden_ratio_approximation(92) = 1.6180339887498949025257388711906969547271728515625.

--------------------------------
End Of Program
--------------------------------
```

This web page was last updated on 02_OCTOBER_2022. The content displayed on this web page is licensed as PUBLIC_DOMAIN intellectual property.

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This web page was last updated on 23_OCTOBER_2022. The content displayed on this web page is licensed as PUBLIC_DOMAIN intellectual property.