Hello i dont really understand what makes a recurence relati

Hello i dont really understand what makes a recurence relation linear. For example:

a(n+1) = 3n^4 + a(n) + a(n-1) + n^(4)a(n-5)

This one isnt homogenous because the term 3n^4 doesnt have a in it right?

It doesnt have constant coefficents.

\'But here is what confuses me, is it linear? because n^4 is before a(n-5) or is it only nonlinear if a has another degree for example (a(n-5)^3

Thanks alot.

Solution

definition

A inear non-homogenous recurrence relationl with constant coefficients is a recurrence relation of the form a(n) =c1a(n-1) + c2a(n-2) + … + cka(n-k)+ f(n), where c1, c2, …, ck are real numbers, and f(n) is a function depending only on n.

The recurrence relation a(n) = c1a(n-1) + c2a(n-2) + … + cka(n-k), is called the associated homogeneous recurrence relation.

your equation

a(n+1) = 3n^4 + a(n) + a(n-1) + n^(4)a(n-5) is inear non-homogenous recurrence relationland its homogenous part is

a(n+1) = a(n) + a(n-1) + n^(4)a(n-5) and its degree is 5

another degree for example (a(n-5)^3 is also 5