Project Euler 2

Problem #2:Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed four million.

Solution: A straight forward implementation in C for this problem can be as shown

#include
#define LIMIT 4000000
int main()
{
 long a=1, b=2, c=0, sum=0;
 sum = b;
 while(c < LIMIT ){
  c = a+b;
  if(c < 1000000)
  if(c%2 == 0)
   sum += c;
  a = b;
  b = c;
 }
 printf("\nAnswer to euler #2 : %ld\n",sum);
}

        
                 This code works fine for our required need, but more optimizations can be done.How? Lets look at the fibo seq.

                                                    1 1 2 3 5 8 13 21 34 55 89 144 ...

                 As we can see,every 3rd term is a even number. Therefore we can change the code accordingly to just sum up every third number.
                 Another approach is we can find a recurrence funtion for this new series i.e. we can have :

F`(n)=4*F`(n-1) + F`(n-2)

It is easy to prove that for Fibonacci series we get:

 F(n)=4*F(n-3)+F(n-6) 

Thus we get a more faster running code, which we eventually need :-)

Enjoy coding :-)