Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients

Question

How would I find sequences that satisfy the following relation?

$$a_{n+2} = -a_{n+1} + 5a_{n}$$

$$\text{given:}\quad a_{0} = 2, a_{1} = 8, \text{ and }(n \ge 0)$$

Answer 1

You can find every sequence satisfying $b_{n+2}=-b_{n+1}+5b_n$ by supposing $b_n=c^n$. This yields: $$ c^{n+2}=-c^{n+1}+5c^n \iff c^n\left(c^2+c-5\right)=0$$ This equation has the three roots $$c_0=0, c_1=\frac{\sqrt{21}-1}{2}, c_2=\frac{-\sqrt{21}-1}{2} $$ So the sequences $b_n=c_1^n$ and $b_n=c_2^n$ satisfy the given relation. And so does an arbitrary linear combination $xc_1^n+yc_2^n$. Now you need to solve $$ xc_1^0+yc_2^0=x+y=2 $$ $$ xc_1^1+yc_2^1=xc_1+yc_2=8 $$ For $x,y$.