i found a general formula in any given set of Fibonacci numbers ,to find the next given even number
we can use the formula
E*4 + Eo
where E is the given even number
Eo is the even number that comes before the given even number
for example :
1,2,3,5,8 to find the even number that occurs after 8
we use the formula E*4 + Eo
ie; 8*4 + 2 = 34
so the next even number in the series is 34
can anybody help verify my results ?
Best Answer
Consecutive Fibonacci numbers are relatively prime, which is fairly easy to establish, so there are no consecutive even Fibonacci numbers.
Thus, if $F_{n}$ is even, then $F_{n-1}$ is odd, $F_{n+1}$ is odd, $F_{n+2}$ is odd, and $F_{n+3}$ is even. So every third Fibonacci number is even. In other words, your formula says for even $F_{n}$, $$ F_{n+3} = 4F_{n} + F_{n-3} $$
Can you see why this formula is true?