Understanding difference equation

Question

I was given an example

$$R_n = R_{n-1} + R_{n-2} $$

This equation is given as an second-order equation.

Why is it so?

Answer 1

The fact that it is second-order refers to the fact that the largest difference in indices is $2$. For example,

$$ R_{n+4}=3R_{n+1}^2+R_n $$

is a fourth-order difference equation and

$$ R_{n+3}=2R_{n+2}\cdot R_{n+1} $$

is a second order difference equation.

If you're familiar with ODEs, the terminology is analogous.

Answer 2

One explanation is that one solves (see Recurrence relation, Wikipedia, under "Solving") the following homogeneous difference equation (or recurrence relation) with constant coefficients

$$a_{n}+Aa_{n-1}+Ba_{n-2}=0,$$

by means of the second degree characteristic equation

$$r^2+Ar+B=0,$$

pretty much as one woud solve a homogeneous second-order linear ordinary differential equation with constant coefficients.