Solve the linear homogeneous recurrence relation with constant coefficients

Question

$$9a_{n} = 6a_{n-1}-a_{n-2}, a_{0}=6, a_{1}=5$$

So

$$x^n = (6x^{n-1}-x^{n-2})\div9$$

thus

$$[x^2 = (6x-1)\div9] \equiv [x^2 - \frac{2}{3}x + \frac{1}{9} = 0], x=\frac{1}{3}$$

also

$$a_{2}=\frac{8}{3}, a_{3}=\frac{31}{27}$$

How do I plug in that x/root to solve for the given recurrence relation?

I tried

$$9a_{n} = 6(\frac{1}{3})^{n-1} - (\frac{1}{3})^{n-2}$$

which for n=1 gives 5... which is correct, for a_1 though, not 9a_1? So that's not right.

Answer 1

Since $a_n=c\left(\frac{1}{3}\right)^n+dn\left(\frac{1}{3}\right)^n$, $a_0=2\implies c=6$ and $a_1=5\implies d=9$, so

$\displaystyle a_n=\frac{6}{3^n}+\frac{9n}{3^n}=\frac{2+3n}{3^{n-1}}$

Answer 2

Here are the steps $$ a_{0}=6 $$ $$ a_{1}=5 $$ $$ 9a_{n} = 6a_{n-1}-a_{n-2} $$ Lets rewrite this recurrence as $$ a_{n} = \frac69a_{n-1}-\frac19a_{n-2} $$ $$ a_{n} = \frac23a_{n-1}-\frac1{3^2}a_{n-2} $$ Now let's multiply by $3^n$ $$ 3^0a_{0}=3^06=6 $$ $$ 3^1a_{1}=3^15=15 $$ $$ 3^na_{n} = 2\cdot 3^{n-1}a_{n-1}-3^{n-2}a_{n-2} $$ Let $S_n=3^na_n$, then $$ S_0=6 $$ $$ S_{1}=15 $$ $$ S_n= 2S_{n-1}-S_{n-2} $$ It follows that $$ S_n= S_1n-S_0(n-1)=9n+6=3(3n+2) $$ Therefore

$$ a_n=\frac{3(3n+2)}{3^n}=3^{1-n}(3n+2) $$