# Even Fibonacci Numbers Sum

Given a limit, find the sum of all the even-valued terms in the Fibonacci sequence below given limit.

The first few terms of Fibonacci Numbers are, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ,… (Even numbers are highlighted).

Examples :

```Input : limit = 8
Output : 10
Explanation : 2 + 8 = 10

Input : limit = 400;
Output : 188.
Explanation : 2 + 8 + 34 + 144 = 188.
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

A simple solution is to iterate through all prime numbers while the next number is less than or equal to given limit. For every number, check if it is even. If the number is even, add it to the result.

An efficient solution is based on the below recursive formula for even Fibonacci Numbers

```Recurrence for Even Fibonacci sequence is:
EFn = 4EFn-1 + EFn-2
with seed values
EF0 = 0 and EF1 = 2.

EFn represents n'th term in Even Fibonacci sequence.
```

Refer this more details of above formula.

So while iterating over Fibonacci numbers, we only generate even Fibonacci numbers.

## C++

 `// Find the sum of all the even-valued terms in ` `// the Fibonacci sequence which do not exceed ` `// given limit. ` `#include ` `using` `namespace` `std; ` ` `  `// Returns sum of even Fibonacci numbers which are ` `// less than or equal to given limit. ` `int` `evenFibSum(``int` `limit) ` `{ ` `    ``if` `(limit < 2) ` `        ``return` `0; ` ` `  `    ``// Initialize first two even prime numbers ` `    ``// and their sum ` `    ``long` `long` `int` `ef1 = 0, ef2 = 2; ` `    ``long` `long` `int` `sum = ef1 + ef2; ` ` `  `    ``// calculating sum of even Fibonacci value ` `    ``while` `(ef2 <= limit) ` `    ``{ ` `        ``// get next even value of Fibonacci sequence ` `        ``long` `long` `int` `ef3 = 4*ef2 + ef1; ` ` `  `        ``// If we go beyond limit, we break loop ` `        ``if` `(ef3 > limit) ` `            ``break``; ` ` `  `        ``// Move to next even number and update sum ` `        ``ef1 = ef2; ` `        ``ef2 = ef3; ` `        ``sum += ef2; ` `    ``} ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `limit = 400; ` `    ``cout << evenFibSum(limit); ` `    ``return` `0; ` `} `

## Java

 `// Find the sum of all the even-valued terms in ` `// the Fibonacci sequence which do not exceed ` `// given limit. ` `import` `java.io.*; ` ` `  `class` `GFG  ` `{ ` `    ``// Returns sum of even Fibonacci numbers which are ` `    ``// less than or equal to given limit. ` `    ``static` `int` `evenFibSum(``int` `limit) ` `    ``{ ` `        ``if` `(limit < ``2``) ` `            ``return` `0``; ` `     `  `        ``// Initialize first two even prime numbers ` `        ``// and their sum ` `        ``long` `ef1 = ``0``, ef2 = ``2``; ` `        ``long` `sum = ef1 + ef2; ` `     `  `        ``// calculating sum of even Fibonacci value ` `        ``while` `(ef2 <= limit) ` `        ``{ ` `            ``// get next even value of Fibonacci sequence ` `            ``long` `ef3 = ``4` `* ef2 + ef1; ` `     `  `            ``// If we go beyond limit, we break loop ` `            ``if` `(ef3 > limit) ` `                ``break``; ` `     `  `            ``// Move to next even number and update sum ` `            ``ef1 = ef2; ` `            ``ef2 = ef3; ` `            ``sum += ef2; ` `        ``} ` `     `  `        ``return``(``int``) sum; ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `main (String[] args)  ` `    ``{ ` `        ``int` `limit = ``400``; ` `        ``System.out.println(evenFibSum(limit)); ` `         `  `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## Python3

 `# Find the sum of all the even-valued  ` `# terms in the Fibonacci sequence which  ` `# do not exceed given limit. ` ` `  `# Returns sum of even Fibonacci numbers which ` `# are less than or equal to given limit. ` `def` `evenFibSum(limit) : ` `    ``if` `(limit < ``2``) : ` `        ``return` `0` ` `  `    ``# Initialize first two even prime numbers ` `    ``# and their sum ` `    ``ef1 ``=` `0` `    ``ef2 ``=` `2` `    ``sm``=` `ef1 ``+` `ef2 ` `     `  `    ``# calculating sum of even Fibonacci value ` `    ``while` `(ef2 <``=` `limit) : ` ` `  `        ``# get next even value of Fibonacci  ` `        ``# sequence ` `        ``ef3 ``=` `4` `*` `ef2 ``+` `ef1 ` ` `  `        ``# If we go beyond limit, we break loop ` `        ``if` `(ef3 > limit) : ` `            ``break` ` `  `        ``# Move to next even number and update ` `        ``# sum ` `        ``ef1 ``=` `ef2 ` `        ``ef2 ``=` `ef3 ` `        ``sm ``=` `sm ``+` `ef2 ` `     `  `    ``return` `sm ` ` `  `# Driver code ` `limit ``=` `400` `print``(evenFibSum(limit)) ` ` `  `# This code is contributed by Nikita Tiwari. `

## C#

 `// C# program to Find the sum of all ` `// the even-valued terms in the  ` `// Fibonacci sequence which do not ` `// exceed given limit.given limit. ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Returns sum of even Fibonacci  ` `    ``// numbers which are less than or ` `    ``// equal to given limit. ` `    ``static` `int` `evenFibSum(``int` `limit) ` `    ``{ ` `        ``if` `(limit < 2) ` `            ``return` `0; ` `     `  `        ``// Initialize first two even ` `        ``// prime numbers and their sum ` `        ``long` `ef1 = 0, ef2 = 2; ` `        ``long` `sum = ef1 + ef2; ` `     `  `        ``// calculating sum of even  ` `        ``// Fibonacci value ` `        ``while` `(ef2 <= limit) ` `        ``{ ` `             `  `            ``// get next even value of  ` `            ``// Fibonacci sequence ` `            ``long` `ef3 = 4 * ef2 + ef1; ` `     `  `            ``// If we go beyond limit, ` `            ``// we break loop ` `            ``if` `(ef3 > limit) ` `                ``break``; ` `     `  `            ``// Move to next even number ` `            ``// and update sum ` `            ``ef1 = ef2; ` `            ``ef2 = ef3; ` `            ``sum += ef2; ` `        ``} ` `     `  `        ``return``(``int``) sum; ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `Main ()  ` `    ``{ ` `        ``int` `limit = 400; ` `        ``Console.Write(evenFibSum(limit)); ` `         `  `    ``} ` `} ` ` `  `// This code is contributed by Nitin Mittal. `

## PHP

 ` ``\$limit``) ` `            ``break``; ` ` `  `        ``// Move to next even number ` `        ``// and update sum ` `        ``\$ef1` `= ``\$ef2``; ` `        ``\$ef2` `= ``\$ef3``; ` `        ``\$sum` `+= ``\$ef2``; ` `    ``} ` ` `  `    ``return` `\$sum``; ` `} ` ` `  `// Driver code ` `\$limit` `= 400; ` `echo``(evenFibSum(``\$limit``)); ` ` `  `// This code is contributed by Ajit. ` `?> `

Output :

```188
```

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