Hello Bryan,
A) We are given the recurrence relation
$$A_{n}=A_{n-1}+6A_{n-2}$$ where $$A_0=3,\,A_1=6$$
The associated characteristic equation is:
$$r^2-r-6=(r+2)(r-6)=0$$
Our characteristic roots are then $$r=-2,\,6$$
Hence, the closed form is:
$$A_n=k_1(-2)^n+k_2(6)^n$$
We may use the initial values to determine the parameters $k_i$:
$$A_0=k_1(-2)^0+k_2(6)^0=k_1+k_2=3$$
$$A_1=k_1(-2)^1+k_2(6)^1=-2k_1+6k_2=6\,\therefore\,-k_1+3k_2=3$$
Solving this 2X2 system, we find:
$$k_1=k_2=\frac{3}{2}$$ and so:
$$A_n=\frac{3}{2}\left((-2)^n+6^n \right)=3\cdot2^{n-1}\left((-1)^n+3^n \right)$$
B) We are given the recurrence relation:
$$A_{n+2}=-4A_{n+1}+5A_{n}$$ where $$A_0=2,\,A_1=8$$
The associated characteristic equation is:
$$r^2+4r-5=(r+5)(r-1)=0$$
Our characteristic roots are then $$r=-5,\,1$$
Hence, the closed form is:
$$A_n=k_1(-5)^n+k_2$$
We may use the initial values to determine the parameters $k_i$:
$$A_0=k_1(-5)^0+k_2=k_1+k_2=2$$
$$A_1=k_1(-5)^1+k_2=-5k_1+k_2=8$$
Solving this 2X2 system, we find:
$$k_1=-1,\,k_2=3$$ and so:
$$A_n=3-(-5)^n$$
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