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I am trying to learn recurrence relations and I want to find the general formula of second-order linear homogeneous relation

$a_n=6a_{n-1}-8a_{n-2}$ where $a_0=1; a_1=4.$

So I started with the characteristic equation and I got this:

$r^2-6r+8=0$

After finding roots I should find a general formula for this relation:

$a_n=\alpha(4^n)+\beta(2^n)$

after plugging given values, I have got an equation:

$a_n=4^n$, and it's incorrect. Can someone help me and explain where I have a mistake? Also is it possible to solve this type of problem with the plug and chug method? Thank you in advance

Nika Chelidze
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Plugging $n=0,1$ in the general solution $a_n = \alpha 4^n + \beta 2^n$ you get the linear system:

$$\begin{cases} \alpha + \beta = 1 \\ 4\alpha + 2\beta = 4\end{cases} \; ,$$

from which you obtain:

$$\alpha = 1,\ \beta = 0 \; .$$

Therefore your solution is correct... Why do you think it is not?

Pacciu
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  • So I checked it a second time and guess it's correct, thanks for your answer. Can you tell me if is it possible to solve this type of relations with the plug and chug method? – Nika Chelidze Oct 24 '20 at 09:54
  • More "plug and chug" than this? I mean, you just "plugged" the guess $a_n = \alpha 4^n + \beta 2^n$ in your initial conditions and, with mild manual effort, you "chugged" the correct values for $\alpha$ and $\beta$... What more can you ask? – Pacciu Oct 25 '20 at 19:26
  • I mean there is method when you calculate $a_{n-1}$ and $a_{n-2}$ and then chug it in equation. After some steps some pattern is visible and you write general formula. I know this method only for first order homogeneous equations, not for this. Can I do this method here too? – Nika Chelidze Oct 26 '20 at 12:03
  • Ah, ok, you mean what I usually call “guessing a formula by examples”… Of course you can use this technique also in the second order setting, but “guessing” the correct formula is quite more complicated than for first order equations. – Pacciu Oct 26 '20 at 15:41
  • Yes I tried I couldn't, I hope someone would teach me – Nika Chelidze Oct 26 '20 at 18:06