Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Find the sum of all the even-valued terms in the sequence which do not exceed four million.

Analysis

The approximate ratio between two consecutive terms in the Fibonacci sequence is the golden ratio (phi, or φ ≈ 1.618034). It can also be demonstrated that every 3rd term of the sequence is even. By combining this knowledge we can calculate the series of even Fibonacci numbers by multiplying the previous even term in the series by the cube of the golden ratio.

Solution

Runs < 1 second in Python.

limit = 4000000
phi_cubed = ((1+5**.5)/2)**3   #golden ratio cubed
 
f = 2
sum = 0
while f < limit:
  sum += f
  f = round(f*phi_cubed)
print "Answewr to PE2 - ",int(sum)

Comments

For a limit of 4,000,000 the loop iterates 10 times.