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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 ?

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    $\begingroup$ Note that modulo $2$, the Fibonacci sequence is periodic with period $3$, and is given by $0, 1, 1, 0, 1, 1, \ldots$, so the even numbers are precisely every third Fibonacci number starting with $F_n = 0$. So, your claim is equivalent to asserting that $F_{n + 6} = 4 F_{n + 3} + F_n$ for all $n$ that are multiples of $3$. In fact, this is true for all $n$. $\endgroup$ – Travis Feb 27 '15 at 6:20
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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?

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$F_n+4F_{n+3}=F_n+F_{n+1}+F_{n+2}+3F_{n+3}$
Now try to combine them using Fibonacci's rule.

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