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?
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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. |
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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. |
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