Considering the Fibonacci sequence satisfies the recurrence relation Fk = Fk-1 + Fk-2, for all integers k ³ 2.

Question

Considering the Fibonacci sequence satisfies the recurrence relation Fk = Fk-1 + Fk-2, for all integers k ³ 2. Whats an equation expressing Fk+3 in terms of Fk+2 and Fk+1

Answer 1

The generating formula can be written as:
F (k+3) = F (k+2) + F (k+1)

Each term is simply the sum of the previous two terms. What you call them is fairly unimportant as long as it is clear they are successive terms, how you generate them, and that the largest term is the latest in the sequence.

Answer 2

Just shift indexes:

f(k) = f(k-1) + f(k-2), for all k >=2. Substituting k by j+3, and assuming k >= 5,

f(j+3) = f(j+2) + f(j+1).