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.
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$\begingroup$ I guess the task is to find the general term, given the first two terms and the recurrence relation. $\endgroup$– Vincenzo OlivaFeb 12, 2015 at 17:50
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$\begingroup$ You mean you want a closed form? Or maybe prove by induction? $\endgroup$– user198044Feb 12, 2015 at 17:51
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$\begingroup$ I just have to find a formula for $x_{n}$ I guess... but how? $\endgroup$– cambelotFeb 12, 2015 at 17:52
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1 Answer
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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 ...