Solve the Recurrence Relation of $A_{n+1} = A_n+A_{n-1}$ for $n\geq2$

Question

I am trying to solve some recurrences for an exam that I found from a past final:

I am given $A_1=1$ and $A_2=3$.

I re-wrote the relation as $A_{n+1} - A_n - A_{n-1} =0$ and found the characteristic polynomial, $x^2-x-x=0$

I solve for x and get $x=0, 2$ from $x(x-2)$

$\longrightarrow$ Solving for A and B from $X_n = A(0^n)+B(2^b)$

I first set $n=0$ and get $0=0+B(2^0) \longrightarrow B=0$

I then set $n=1$ and get $1=0+B(2^1) \longrightarrow 2B=1$

So $B=0$ and $B=1/2$

I am not sure where to go from here, or if I made a mistake since I have $ 2$ different values for $B$.

Some help would be much appreciated!

Thanks

Answer 1

The characteristic polynomial of the recursion $$A_{n+2} - A_{n+1} - A_n = 0$$

is

$$x^2 - x - 1 $$

This should fix your problems.