Find the Nth term of the series where each term f[i] = f[i – 1] – f[i – 2]

Given three integers X, Y and N, the task is to find the N^th term of the series f[i] = f[i – 1] – f[i – 2], i > 1 where f[0] = X and f[1] = Y.
Examples:

Input: X = 2, Y = 3, N = 3
Output: -2
The series will be 2 3 1 -2 -3 -1 2 and f[3] = -2
Input: X = 3, Y = 7, N = 8
Output: 4

Approach: An important observation here is that there will be atmost 6 distinct terms, before the sequence starts repeating itself. So, find the first 6 terms of the series and then the N^th term would be same as the (N % 6)^th term.
Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <bits/stdc++.h>

using namespace std;
 
// Function to return the Nth term
// of the given series

int findNthTerm(int x, int y, int n)
{

    int f[6];
 
    // First and second term of the series

    f[0] = x;

    f[1] = y;
 
    // Find first 6 terms

    for (int i = 2; i <= 5; i++)

        f[i] = f[i - 1] - f[i - 2];
 
    // Return the Nth term

    return f[n % 6];
}
 
// Driver code

int main()
{

    int x = 2, y = 3, n = 3;

    cout << findNthTerm(x, y, n);
 
    return 0;
}

Java

// Java implementation of the approach

import java.io.*; 
 
class GFG 
{ 

    // Function to find the nth term of series 

    static int findNthTerm(int x, int y, int n) 

    {     

        int[] f = new int[6];

        f[0] = x;

        f[1] = y;

        // Loop to add numbers 

        for (int i = 2; i <= 5; i++) 

            f[i] = f[i - 1] - f[i - 2]; 

        return f[n % 6]; 

    } 
 
    // Driver code 

    public static void main(String args[]) 

    { 

        int x = 2, y = 3, n = 3; 

        System.out.println(findNthTerm(x, y, n)); 

    } 
} 
 
// This code is contributed by mohit kumar 29

Python3

# Python3 implementation of the approach 
 
# Function to return the Nth term 
# of the given series 

def findNthTerm(x, y, n): 
 
    f = [0] * 6
 
    # First and second term of 

    # the series 

    f[0] = x 

    f[1] = y 
 
    # Find first 6 terms 

    for i in range(2, 6): 

        f[i] = f[i - 1] - f[i - 2] 
 
    # Return the Nth term 

    return f[n % 6] 
 
# Driver code 

if __name__ == "__main__":
 
    x, y, n = 2, 3, 3

    print(findNthTerm(x, y, n))
 
# This code is contributed by
# Rituraj Jain

// C# implementation of the approach 

using System;
 
class GFG 
{ 

    // Function to find the nth term of series 

    static int findNthTerm(int x, int y, int n) 

    { 

        int[] f = new int[6]; 

        f[0] = x; 

        f[1] = y; 

        // Loop to add numbers 

        for (int i = 2; i <= 5; i++) 

            f[i] = f[i - 1] - f[i - 2]; 

        return f[n % 6]; 

    } 
 
    // Driver code 

    public static void Main() 

    { 

        int x = 2, y = 3, n = 3; 

        Console.WriteLine(findNthTerm(x, y, n)); 

    } 
} 
 
// This code is contributed by Ryuga

Javascript

<script>
 
// JavaScript implementation of the approach 
 
    // Function to return the Nth term 

    // of the given series 

    function findNthTerm(x, y, n) 

    { 

        let f = new Array(6); 

        // First and second term of the series 

        f[0] = x; 

        f[1] = y; 

        // Find first 6 terms 

        for (let i = 2; i <= 5; i++) 

            f[i] = f[i - 1] - f[i - 2]; 

        // Return the Nth term 

        return f[n % 6]; 

    } 

    // Driver code 
 
    let x = 2, y = 3, n = 3; 

    document.write(findNthTerm(x, y, n)); 

// This code is contributed by Surbhi Tyagi
 
</script>

PHP

<?php
 
//PHP implementation of the approach
// Function to find the nth term of series 

function findNthTerm($x, $y, $n)
{ 

    $f = array(6);

    $f[0] = $x;

    $f[1] = $y;

    // Loop to add numbers 

    for ($i = 2; $i <= 5; $i++) 

        $f[$i] = $f[$i - 1] - $f[$i - 2]; 

    return $f[$n % 6]; 
} 
 
// Driver code 

$x = 2; $y = 3; $n = 3; 

echo(findNthTerm($x, $y, $n)); 
 
// This code is contributed by Code_Mech.
?>

Output

-2

Time Complexity: O(1) Since no loop is used the algorithm takes up constant time to perform the operations
Auxiliary Space: O(1) Since no extra array is used so the space taken by the algorithm is constant

Method 2 (Space Optimized Method 2) :
We can optimize the space used in method 2 by storing the previous two numbers only because that is all we need to get the next sequence number in the series.

Steps :

1.Define a function "fib" that takes three integer parameters: x, y, and n.
2.Initialize variables a and b to x and y, respectively, and declare a variable c.
3.If n is equal to zero, return the value of a.
4.For i = 2 to n, calculate the next term of the Fibonacci series using the space-optimized method (c = b - a, a = b, b = c).
5.Return the value of b.
6.In the main function, initialize variables x, y, and n, and call the "fib" function with these values.
7.Print the result.

C++

// Fibonacci Series using Space Optimized Method
#include <bits/stdc++.h>

using namespace std;
 
// Function to generate Fibonacci series using space-optimized method

int fib(int x, int y, int n)
{

    // Initialize variables a and b to x and y, respectively, and declare a variable c and loop variable i.

    int a = x, b = y, c, i;
 
    // If n is equal to zero, return the value of a.

    if (n == 0)

        return a;
 
    // For i = 2 to n, calculate the next term of the Fibonacci series using the space-optimized method (c = b - a, a = b, b = c).

    for (i = 2; i <= n; i++) {

        c = b - a;

        a = b;

        b = c;

    }
 
    // Return the value of b, which represents the nth term of the Fibonacci series.

    return b;
}
 
// Driver code

int main()
{

    // Initialize variables x, y, and n.

    int x = 2, y = 3;

    int n = 3;
 
    // Call the "fib" function with initial values for x, y, and n and print the result.

    cout << fib(2, 3, 3);
 
    return 0;
}

Java

// Fibonacci Series using Space Optimized Method

import java.io.*;
 
class GFG {

    // Function to generate Fibonacci series using

    // space-optimized method

    public static int fib(int x, int y, int n)

    {

        // Initialize variables a and b to x and y,

        // respectively, and declare a variable c and loop

        // variable i.

        int a = x, b = y, c, i;
 
        // If n is equal to zero, return the value of a.

        if (n == 0)

            return a;
 
        // For i = 2 to n, calculate the next term of the

        // Fibonacci series using the space-optimized method

        // (c = b - a, a = b, b = c).

        for (i = 2; i <= n; i++) {

            c = b - a;

            a = b;

            b = c;

        }
 
        // Return the value of b, which represents the nth

        // term of the Fibonacci series.

        return b;

    }
 
    // Driver code

    public static void main(String[] args)

    {

        // Initialize variables x, y, and n.

        int x = 2, y = 3;

        int n = 3;
 
        // Call the "fib" function with initial values for

        // x, y, and n and print the result.

        System.out.println(fib(2, 3, 3));

    }
}
// This code is contributed by Taranpreet Singh

Python3

# Fibonacci Series using Space Optimized Method
 
# Function to generate Fibonacci series using space-optimized method

def fib(x, y, n):

    # Initialize variables a and b to x and y,

    # respectively, and declare a variable c and loop variable i.

    a, b = x, y

    c = 0
 
    # If n is equal to zero, return the value of a.

    if n == 0:

        return a
 
    # For i = 2 to n, calculate the next term of the 

    # Fibonacci series using the space-optimized 

    # method (c = b - a, a = b, b = c).

    for i in range(2, n + 1):

        c = b - a

        a = b

        b = c
 
    # Return the value of b, which 

    # represents the nth term of the Fibonacci series.

    return b
 
# Driver code

def main():

    # Initialize variables x, y, and n.

    x = 2

    y = 3

    n = 3
 
    # Call the "fib" function with initial values for x, y, and n and print the result.

    print(fib(x, y, n))
 
if __name__ == "__main__":

    main()
# This code is contributed by Dwaipayan Bandyopadhyay

using System;
 
class FibonacciSeries
{

    // Function to generate Fibonacci series using space-optimized method

    static int Fib(int x, int y, int n)

    {

        // Initialize variables a and b to x and y, respectively, and declare a variable c and loop variable i.

        int a = x, b = y, c;
 
        // If n is equal to zero, return the value of a.

        if (n == 0)

            return a;
 
        // For i = 2 to n, calculate the next term of the Fibonacci series using the space-optimized method (c = b - a, a = b, b = c).

        for (int i = 2; i <= n; i++)

        {

            c = b - a;

            a = b;

            b = c;

        }
 
        // Return the value of b, which represents the nth term of the Fibonacci series.

        return b;

    }
 
    // Driver code

    static void Main()

    {

        // Call the "Fib" function with initial values for x, y, and n and print the result.

        Console.WriteLine(Fib(2, 3, 3));

    }
}

Javascript

// Nikunj Sonigara
 
// Function to generate Fibonacci series using space-optimized method

function fib(x, y, n) {

    // Initialize variables a and b to x and y, respectively, and declare a variable c.

    let a = x;

    let b = y;

    let c;
 
    // If n is equal to zero, return the value of a.

    if (n === 0) {

        return a;

    }
 
    // For i = 2 to n, calculate the next term of the Fibonacci 

    // series using the space-optimized method (c = b - a, a = b, b = c).

    for (let i = 2; i <= n; i++) {

        c = b - a;

        a = b;

        b = c;

    }
 
    // Return the value of b, which represents the nth term of the Fibonacci series.

    return b;
}
 
// Driver code

function main() {

    // Initialize variables x, y, and n.

    const x = 2;

    const y = 3;

    const n = 3;
 
    // Call the "fib" function with initial values for x, y, and n and print the result.

    console.log(fib(2, 3, 3));
}
 
// Call the "main" function to start the program.
main();

Output

-2

Time Complexity: O(n)
Auxiliary Space: O(1)

Article Tags :

DSA

Mathematical

Fibonacci

math