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I understand how to solve a recurrence relation of the form:

$ U_n = 7U_{n-1} + 18U_{n-2} $

which becomes

$ U_n - 7U_{n-1} - 18U_{n-2} = 0 $

and to characteristic form

$ x^2 - 7x - 18 = 0 $

However i do not understand what to do when there are extras added not of the form "$U_{n-a}$" for example:

$ U_n = 3U_{n-1} - 3U_{n-2} + U_{n-3} + 17 $
or
$ U_n = 7U_{n-1} + 18U_{n-2} + 3^n $

i assume i follow the procedure of

$ U_n - 3U_{n-1} + 3U_{n-2} - U_{n-3} - 17 = 0 $

but from here i do not know how to continue

1 Answers1

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If you have a recurrence like $$U_n=aU_{n-1}+bU_{n-2}+c$$ subtract the $n-1$ case from it to give $$U_n-U_{n-1}=a(U_{n-1}-U_{n-2})+b(U_{n-2}-U_{n-3})$$ which can be written in the form $$U_n=a'U_{n-1}+b'U_{n-2}+c'U_{n-3}.$$

Or if you have $$U_n=aU_{n-1}+bU_{n-2}+c^n$$ subtract $c$ times the $n-1$ case from it to give $$U_n-cU_{n-1}=a(U_{n-1}-cU_{n-2})+b(U_{n-2}-cU_{n-3})$$ etc.

Angina Seng
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  • I will add that it is an example of the transformation of an "affine issue", i.e. with a "translational component" (what you call an "extra") into a "vectorial issue" (without this component). – Jean Marie May 07 '17 at 06:26