Finding sequence with $a_n - a_{n+1} = a_{n+2}$ [closed]

Solution 1:

To study sequences type : $$x_{n + 1} = a x_{n + 1} + b x_n$$ consider the associated equation : $$t^2 = a t + b$$ If the equation has :

1. Two different solutions $r$ and $s$ then : $$\exists A, B \in \mathbb{R}, \forall n \in \mathbb{N}, x_n = A r^n + B s^n$$
2. One solution $r$ then : $$\exists A, B \in \mathbb{R}, \forall n \in \mathbb{N}, x_n = (A n + B) r^n$$
3. Two different complexe conjugate solutions $r e^{i \theta}$ and $r e^{-i \theta}$ then : $$\exists A, B \in \mathbb{R}, \forall n \in \mathbb{N}, x_n = r^n (A \cos n \theta + B \sin n \theta)$$

Solution 2:

Let $f_n:=(-1)^na_n$. We have

$$f_{n+2}=f_{n+1}+f_n,$$ which is the Fibonacci recurrence. From Binet's formula, we recall that the Fibonacci numbers are the sum of a growing exponential $\phi^n$ and and alternating one, $(-\phi)^{-n}$.

Hence,

$$a_n=\phi^{-n}$$ can do.