Find an explicit formula for $s_n$ if $s_0,s_1,s_2....$ is a sequence satisfying the given recurrence relation and initial conditions. I'm trying to figure out how to finish the formula.
$s_n = -8s_{n-1} - 15s_{n-2}, s_0=2, s_1=2$
What I did was
$x^2=-8x -15$ = $x^2+8x+15$ =$(x+5)(x+3)$
$s_0=c_1(-5)^0+c_2(-3)^0 = c_1+c_2$
$s_1=c_1(-5)^1+c_2(-3)^1 = (-5c_1)+(-3c_2)$
2 Answers
You have done all the hard work.
$$s_n=c_1(-5)^n+c_2(-3)^n$$
Just solve for the arbitrary constants $c_1,c_2$ from the two linear simultaneous equations
$c_1+c_2=s_0=2\ \ \ \ (1)$
$c_1(-5)+c_2(-3)=s_1=2\ \ \ \ (2)$
You look fine so far. What you have now are the equations $s_0=c_1+c_2$ and $s_1=-5c_1-3c_2$. This is a system of linear equations, and can be solved for $c_1$ and $c_2$ in terms of $s_0$ and $s_1$. Solve this system, and you're essentially done.
