With Equation 15.31, you saw recursion of the form f(n – c) in a recurrence relation, where c is a constant, lead to division of the form n/c in the closed form. This illustrates a general relationship between expressions in a recurrence relation and in its closed form, and can help you guess closed forms for many recurrences. Prove the following examples:
1. Recurrence relations of the form:
where c is a constant, have closed forms of the form: f(n) = 2⌊n/c⌋ – 1
2. Recurrence relations of the form:
where c is a constant, have closed forms of the form:
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