Problem with solving the recurrence relation $a_n=a_{n-1}+6a_{n-2}+30$ for $n\geq2$, $a_0=0$, $a_1=-10$

Question

My task: $a_n=a_{n-1}+6a_{n-2}+30$ for $n\geq2$, $a_0=0$, $a_1=-10$

My solution $x^{2}-x-6$

$\Delta=25$

$x1=-2 $

$x2=3$

So I am gonna use following formula: $a_n=ar^{n}+br^{n}$

$a_n=a*(-2)^{n}+b*3^{n}$

$a_0=0=a+b$

$a_1=-10=-2a+3b$

$b=-2$

$a=2$

$a_n=2*(-2)^{n}-2*3^{n}+30$

I calculate $a_2$

$a_2=-10+0+30=20$

Which is correct with above formula

$a_2=2*(-2)^{2}-2*3^{2}+30=20$

but for $a_3$ and above results are not matching, am I doing something wrong here?

Answer 1

The $+30$ is the problem. Find $c$ such that $b_n:=a_n+c\implies b_n=b_{n-1}+6b_{n-2}$, then use your usual techniques.

Edit to add detail:

$$b_n=b_{n-1}-c+6(b_{n-2}-c)+30+c=b_{n-1}+6b_{n-2}+30-6c$$ Set $c=5$ so $$b_n=b_{n-1}+6b_{n-2}\implies\exists a,\,b:\,b_n=a(-2)^n+b3^n.$$From $b_0=5,\,b_1=-5$, you can find $a,\,b$. Then $a_n=b_n-5$.

Answer 2

Since for all $n$ we have $$a_n-a_{n-1}-6a_{n-2}=30$$ we have also

$$a_{n-1}-a_{n-2}-6a_{n-3}=30$$

so $$a_n-a_{n-1}-6a_{n-2}=a_{n-1}-a_{n-2}-6a_{n-3}$$

and thus we get l.r. :

$$a_n-2a_{n-1}-5a_{n-2}+6a_{n-3}=0$$

and so on...