GOLDEN_RATIO_APPROXIMATION_15_NOVEMBER_2022_COPY


GOLDEN_RATIO_APPROXIMATION_15_NOVEMBER_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 ā€“ 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


C++ source file: https://github.com/karlinarayberinger/karlina_object_ultimate_starter_pack/blob/main/golden_ratio_approximation.cpp

plain-text file: https://github.com/karlinarayberinger/karlina_object_ultimate_starter_pack/blob/main/golden_ratio_approximation_output.txt


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


C++ source file: https://github.com/karlinarayberinger/karlina_object_ultimate_starter_pack/blob/main/golden_ratio_approximation.cpp

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
 * license: PUBLIC_DOMAIN
 */

/* 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


plain-text file: https://github.com/karlinarayberinger/karlina_object_ultimate_starter_pack/blob/main/golden_ratio_approximation_output.txt


--------------------------------
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 15_NOVEMBER_2022. The content displayed on this web page is licensed as PUBLIC_DOMAIN intellectual property.