Find $r$, given that $F_r= 2F_{101}+F_{100}$

Question

Find $r$, given that $F_r= 2F_{101}+F_{100}$.

We know that the recurrence relation for the Fibonacci sequence is $F_n= F_{n-1}+F_{n-2}$ and that $F_0 = F_1 = 1$, but how to proceed further?

Answer 1

$$F_{103}=F_{102}+F_{101},\quad F_{102}=F_{101}+F_{100}\quad \implies \quad F_{103}=2F_{101}+F_{100}$$

Answer 2

You can repeat the relation,

$$ F_{n} = F_{n-1} + F_{n-2} $$ $$ F_{n+1} = F_{n} + F_{n-1} $$ $$ \rightarrow F_{n+1} = [F_{n-1} + F_{n-2}] + F_{n-1} $$ $$ F_{n+1} = 2 \cdot F_{n-1} + F_{n-2} $$

You could do this as many times as you like and get to the expression of any number in the sequence in terms of the 2 first ones.