Using the definition of the Fibonacci recurrence relation. $F_{2j} + F_{2j+1} $ to $F_{2k+2}$

Question

How do I go from $F_{2j} + F_{2j+1} $ to $F_{2k+2}$.

Answer 1

$F_{2k+2}=F_{2k+1}+F_{2k}$

$=2F_{2k}+F_{2k-1}$

$=3F_{2k-1}+2F_{2k-2}$

$=5F_{2k-2}+3_{2k-3}$

The pattern is pretty obvious from here in terms of Fibonacci Numbers