A linear recurrence relation with constant coefficients expresses each term as a linear combination of earlier terms.

For example,

$$a_n=3a_{n-1}-2a_{n-2}.$$

To look for solutions of the form $a_n=r^n$, substitute into the recurrence:

$$r^n=3r^{n-1}-2r^{n-2}.$$

For $r\ne0$, this gives the characteristic equation

$$r^2-3r+2=0,$$

whose roots are $1$ and $2$.

With distinct roots $r_1,r_2$, solutions have the form

$$a_n=A r_1^n+B r_2^n,$$

where the initial conditions determine $A$ and $B$.

Repeated roots require an additional polynomial factor in $n$. This method turns many discrete evolution rules into explicit formulas that expose long-term growth.