Let $x_{0}=1$ and $x_{1}=-1$ For $n\geq0$ inductively define $x_{n+2}=x_{n+1}+6x_{n}$

Question

I am not so sure how to do this problem and would like some help here. How would you induct a relation given this information here? I mean I know what induction means but I'm not so sure what I'm being asked to do.

Answer 1

Another way to solve it :
you can solve it by Characteristic equation $$x_{n+2}=x_{n+1}+6x_n\\r^2=r+6\\r^2-r-6=0\\(r-3)(r+2)=0\\r=3,-2\\$$so $x_n$ must be like this $$x_{n}=c_1r_1^n+c_2r_2^n\\x_n=c_1(3)^n+c_2(-2)^n\\$$now apply $x_1=-1,x_0=1$ to find $c_1,c_2$ at the end $$ x_{n}=\frac{1}{5}(3)^n+\frac{4}{5}(-2)^n$$ you can easily see $x_2=5$ and so on ...