Suppose a recurrence relation
an=an−1+20an−2
where a1=9 and a2=189
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.
**If the explicit formula is in the form of Formula 2, consider p > q.
Determine p and q
Answer:
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
Therefore, and
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