Simple recession formula

Question

I am asked to find a closed formula (no dependence on other parts of the sequence) for the following formula: $x_n = 3x_{n-1} - x_{n-2}$ where $x_1=3$, $x_2=8$ and $x_3=21$

How do I go about it?

Answer 1

Hint

The characteristic function is $$\lambda ^2=3\lambda -1$$which has two distinct roots $\lambda_1$ and $\lambda_2$. So the general form is $$x_n=a\lambda_1^n+b\lambda_2^n$$

Answer 2

You could use matrices.

Let $A_n=\pmatrix{x_{n+1}\\x_{n}}.$

Then $A_{n}=\pmatrix{3&-1\\1&0}^{n-1}\pmatrix{8\\3}.$