# Sum of Fibonacci numbers at even indexes upto N terms

• Last Updated : 25 Aug, 2022

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.

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 alternating 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 alternating 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 alternating 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 alternating 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

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## Javascript

 ``

Output:

`Even indexed Fibonacci Sum upto 8 terms: 1596`

Time Complexity: O(n)

Auxiliary Space: O(n)

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`

## Javascript

 ``

Output:

`Even indexed Fibonacci Sum upto 8 terms: 1596`

Time Complexity: O(logn)

Auxiliary Space: O(logn)

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