MATH307 April 2009
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Question 07 (a)
Given the recurrence relation with the initial condition and ,
(a) Solve the recurrence relation. (Give a general scalar expression for in terms of n, a, and b).
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We can start by finding the first few values of this relation. With these, we can guess at a scalar expression for and then use induction to prove that it is correct. By inspection, we come up with the following:
Proof: : : : , which is what we found using the recurrence.
Assume that for , the expression is true.
We now want to prove that the expression is true for . Using the recurrence relation and our induction hypothesis, we find that , which is the expression we had come up with. So for all integers n, .