Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Examples: Typical ones include
• P n = 1.15$P n-1 is degree 1.
• f n = f n-1 + f n-2 is degree 2.
• a n = a n-5 is degree 5.
Examples: Ones that fail the definition include
• a n = a n-1 + a2 n-2
is nonlinear.
• H n = 2H n-1 + 1 is nonhomogeneous.
• B n = nB n-1 is variable coefficient.
We will get to nonhomogeneous recurrence relations shortly.
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Solving a recurrence relation usually assumes that the solution has the form
a n = r n ,
where r,C, if and only if
r n = c 1 r n-1 + c 2 r n-2 + … + c n-k r n-k .
Dividing both sides by r n-k to simplify things, we get
Definition: The characteristic equation is
r k 4 c 1 r k-1 4 c 2 r k-2 4 … 4 c n-k = 0.
Then {a n } with a n = r n is a solution if and only if r is a solution to the
characteristic equation. The proof is quite involved.
The n = 2 case is much easier to understand, yet still multiple cases.
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