Find the characteristic root of the recurrence relation an=anà ƒ ƒ ¢ ˆ ’1+2anà ƒ ƒ ¢ ˆ ’2.

Let us find the characteristic roots of the recurrence relation $a_n=a_{n-1}+2a_{n-2}.$ The characteristic equation $k^2=k+2$ of the recurrence relation is equivalent to $k^2-k-2=0,$ and hence to $(k+1)(k-2)=0.$ It follows that the characteristic roots of the recurrence relation are $k_1=-1$ and $k_2=2.$