Question

Show that the explicit sequence {yn}∞n=0 such that yn = A(2n )+ B(-1)n for any nonzero

constants A and B is the solution of the recurrence relation

yn = yn-1 + 2yn-2 for n >1.

Answer 1

Let us solve the reccurence relation

$y_n = y_{n-1 }+ 2y_{n-2}.$

Its characteristic equation $k^2=k+2$ is equivalent to $k^2-k-2=0,$ and hence to $(k-2)(k+1)=0.$

Therefore, the last equation has the roots $k_1=2,\ k_2=-1.$

We conclude that the general solution of the reccurence relation is of the form:

$y_n=A\cdot 2^n+B\cdot(-1)^n.$