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 Nth 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] = -2Input: 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 Nth 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#
// 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 |
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. ?> |
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> |
-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; } |
-2
Time Complexity: O(n)
Auxiliary Space: O(1)
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