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.
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Let us solve the reccurence relation
Its characteristic equation is equivalent to and hence to
Therefore, the last equation has the roots
We conclude that the general solution of the reccurence relation is of the form: