Suppose a recurrence relation an=2an−1−an−2 where a1=7 and a2=10 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where x and y are roots of the characteristic equation. Determine p and q
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Let us solve the characteristic equation of the recurrence relation which is equivalent to It follows that the characteristic equation is equivalent to and hence has the roots It follows that the solution of the recurrence equation is Since and we conclude that and Therefore, and