Solution to Show that the explicit sequence {yn}∞n=0 such that yn = A(2n )+ B(-1)n for any … - Sikademy
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Archangel Macsika

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

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.



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