# Sum of Fibonacci numbers at even indexes upto N terms

Given a positive integer N, the task is to find the value of F2 + F4 + F6 +………+ F2n upto N terms where Fi denotes the i-th Fibonacci number.

The Fibonacci numbers are the numbers in the following integer sequence.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ……

Examples:

Input: n = 5
Output: 88
N = 5, So the fibonacci series will be generated from 0th term upto 10th term:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Sum of elements at even indexes = 0 + 1 + 3 + 8 + 21 + 55

Input: n = 8
Output: 1596
0 + 1 + 3 + 8 + 21 + 55 + 144 + 377 + 987 = 1596.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Method-1: This method includes solving the problem directly by finding all Fibonacci numbers till 2n and adding up the only the even indices. But this will require O(n) time complexity.

Below is the implementation of the above approach:

## C++

 // C++ Program to find  sum // of even-indiced Fibonacci numbers #include using namespace std;    // Computes value of first fibonacci numbers // and stores the even-indexed sum int calculateEvenSum(int n) {     if (n <= 0)         return 0;        int fibo[2 * n + 1];     fibo[0] = 0, fibo[1] = 1;        // Initialize result     int sum = 0;        // Add remaining terms     for (int i = 2; i <= 2 * n; i++) {         fibo[i] = fibo[i - 1] + fibo[i - 2];            // For even indices         if (i % 2 == 0)             sum += fibo[i];     }        // Return the alternting sum     return sum; }    // Driver program to test above function int main() {        // Get n     int n = 8;        // Find the even-indiced sum     cout << "Even indexed Fibonacci Sum upto "          << n << " terms: "          << calculateEvenSum(n) << endl;        return 0; }

## Java

 // Java Program to find sum // of even-indiced Fibonacci numbers    import java.io.*;    class GFG {       // Computes value of first fibonacci numbers // and stores the even-indexed sum static int calculateEvenSum(int n) {     if (n <= 0)         return 0;        int fibo[] = new int[2 * n + 1];     fibo[0] = 0; fibo[1] = 1;        // Initialize result     int sum = 0;        // Add remaining terms     for (int i = 2; i <= 2 * n; i++) {         fibo[i] = fibo[i - 1] + fibo[i - 2];            // For even indices         if (i % 2 == 0)             sum += fibo[i];     }        // Return the alternting sum     return sum; }    // Driver program      public static void main (String[] args) {             // Get n     int n = 8;        // Find the even-indiced sum     System.out.println("Even indexed Fibonacci Sum upto "         + n + " terms: "+         + calculateEvenSum(n));        } }    // This code is contributed // by shs

## Python 3

 # Python3 Program to find sum  # of even-indiced Fibonacci numbers     # Computes value of first fibonacci # numbers and stores the even-indexed sum  def calculateEvenSum(n) :        if n <= 0 :         return 0        fibo = [0] * (2 * n + 1)     fibo[0] , fibo[1] = 0 , 1        # Initialize result     sum = 0        # Add remaining terms      for i in range(2, 2 * n + 1) :            fibo[i] = fibo[i - 1] + fibo[i - 2]            # For even indices          if i % 2 == 0 :             sum += fibo[i]        # Return the alternting sum      return sum    # Driver code if __name__ == "__main__" :        # Get n      n = 8        # Find the even-indiced sum      print("Even indexed Fibonacci Sum upto",            n, "terms:", calculateEvenSum(n))    # This code is contributed  # by ANKITRAI1

## C#

 // C# Program to find sum of  // even-indiced Fibonacci numbers  using System;    class GFG {        // Computes value of first fibonacci // numbers and stores the even-indexed sum  static int calculateEvenSum(int n)  {      if (n <= 0)          return 0;         int []fibo = new int[2 * n + 1];      fibo[0] = 0; fibo[1] = 1;         // Initialize result      int sum = 0;         // Add remaining terms      for (int i = 2; i <= 2 * n; i++)     {          fibo[i] = fibo[i - 1] +                    fibo[i - 2];             // For even indices          if (i % 2 == 0)              sum += fibo[i];      }         // Return the alternting sum      return sum;  }     // Driver Code  static public void Main () {     // Get n      int n = 8;             // Find the even-indiced sum      Console.WriteLine("Even indexed Fibonacci Sum upto " +                     n + " terms: " + calculateEvenSum(n));  }  }     // This code is contributed // by Sach_Code

## PHP



Output:

Even indexed Fibonacci Sum upto 8 terms: 1596

Method-2:

It can be clearly seen that the required sum can be obtained thus:
2 ( F2 + F4 + F6 +………+ F2n ) = (F1 + F2 + F3 + F4 +………+ F2n) – (F1 – F2 + F3 – F4 +………+ F2n)

Now the first term can be obtained if we put 2n instead of n in the formula given here.

Thus F1 + F2 + F3 + F4 +………+ F2n = F2n+2 – 1.

The second term can also be found if we put 2n instead of n in the formula given here

Thus, F1 – F2 + F3 – F4 +………- F2n = 1 + (-1)2n+1F2n-1 = 1 – F2n-1.

So, 2 ( F2 + F4 + F6 +………+ F2n)
= F2n+2 – 1 – 1 + F2n-1
= F2n+2 + F2n-1 – 2
= F2n + F2n+1 + F2n+1 – F2n – 2
= 2 ( F2n+1 -1)
Hence, ( F2 + F4 + F6 +………+ F2n) = F2n+1 -1 .

So in order to find the required sum, the task is to find only F2n+1 which requires O(log n) time.( Refer to method 5 or method 6 in this article.

Below is the implementation of the above approach:

## C++

 // C++ Program to find even indexed Fibonacci Sum in // O(Log n) time.    #include using namespace std;    const int MAX = 1000;    // Create an array for memoization int f[MAX] = { 0 };    // Returns n'th Fibonacci number // using table f[] int fib(int n) {     // Base cases     if (n == 0)         return 0;     if (n == 1 || n == 2)         return (f[n] = 1);        // If fib(n) is already computed     if (f[n])         return f[n];        int k = (n & 1) ? (n + 1) / 2 : n / 2;        // Applying above formula [Note value n&1 is 1     // if n is odd, else 0].     f[n] = (n & 1) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1))                    : (2 * fib(k - 1) + fib(k)) * fib(k);        return f[n]; }    // Computes value of even-indexed  Fibonacci Sum int calculateEvenSum(int n) {     return (fib(2 * n + 1) - 1); }    // Driver program to test above function int main() {     // Get n     int n = 8;        // Find the alternating sum     cout << "Even indexed Fibonacci Sum upto "          << n << " terms: "          << calculateEvenSum(n) << endl;        return 0; }

## Java

 // Java Program to find even indexed Fibonacci Sum in  // O(Log n) time.     class GFG {        static int MAX = 1000;       // Create an array for memoization      static int f[] = new int[MAX];    // Returns n'th Fibonacci number  // using table f[]      static int fib(int n) {         // Base cases          if (n == 0) {             return 0;         }         if (n == 1 || n == 2) {             return (f[n] = 1);         }            // If fib(n) is already computed          if (f[n] == 1) {             return f[n];         }            int k = (n % 2 == 1) ? (n + 1) / 2 : n / 2;            // Applying above formula [Note value n&1 is 1          // if n is odd, else 0].          f[n] = (n % 2 == 1) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1))                 : (2 * fib(k - 1) + fib(k)) * fib(k);            return f[n];     }    // Computes value of even-indexed Fibonacci Sum      static int calculateEvenSum(int n) {         return (fib(2 * n + 1) - 1);     }    // Driver program to test above function      public static void main(String[] args) {         // Get n          int n = 8;            // Find the alternating sum          System.out.println("Even indexed Fibonacci Sum upto "                 + n + " terms: "                 + calculateEvenSum(n));     } } // This code is contributed by Rajput-Ji

## Python3

 # Python3 Program to find even indexed  # Fibonacci Sum in O(Log n) time. MAX = 1000;    # Create an array for memoization f = [0] * MAX;    # Returns n'th Fibonacci number # using table f[] def fib(n):            # Base cases     if (n == 0):         return 0;     if (n == 1 or n == 2):         f[n] = 1;         return f[n];        # If fib(n) is already computed     if (f[n]):         return f[n];        k = (n + 1) // 2 if (n % 2 == 1) else n // 2;        # Applying above formula [Note value n&1 is 1     # if n is odd, else 0].     f[n] = (fib(k) * fib(k) + fib(k - 1) * fib(k - 1)) \     if (n % 2 == 1) else (2 * fib(k - 1) + fib(k)) * fib(k);        return f[n];    # Computes value of even-indexed Fibonacci Sum def calculateEvenSum(n):     return (fib(2 * n + 1) - 1);    # Driver Code if __name__ == '__main__':            # Get n     n = 8;        # Find the alternating sum     print("Even indexed Fibonacci Sum upto",            n, "terms:", calculateEvenSum(n));    # This code is contributed by PrinciRaj1992

## C#

 // C# Program to find even indexed Fibonacci Sum in  // O(Log n) time.  using System;    class GFG {        static int MAX = 1000;        // Create an array for memoization      static int []f = new int[MAX];        // Returns n'th Fibonacci number      // using table f[]      static int fib(int n)      {         // Base cases          if (n == 0)          {             return 0;         }         if (n == 1 || n == 2)          {             return (f[n] = 1);         }            // If fib(n) is already computed          if (f[n] == 1)          {             return f[n];         }            int k = (n % 2 == 1) ? (n + 1) / 2 : n / 2;            // Applying above formula [Note value n&1 is 1          // if n is odd, else 0].          f[n] = (n % 2 == 1) ? (fib(k) * fib(k) +                                  fib(k - 1) * fib(k - 1))                 : (2 * fib(k - 1) + fib(k)) * fib(k);            return f[n];     }        // Computes value of even-indexed Fibonacci Sum      static int calculateEvenSum(int n)      {         return (fib(2 * n + 1) - 1);     }        // Driver code      public static void Main()      {         // Get n          int n = 8;            // Find the alternating sum          Console.WriteLine("Even indexed Fibonacci Sum upto "                 + n + " terms: "                 + calculateEvenSum(n));     } }    //This code is contributed by 29AjayKumar

Output:

Even indexed Fibonacci Sum upto 8 terms: 1596

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