(5.4.21) Strong Induction Proof of Recursive Sequence Using Floor and Ceiling Functions

This presentation focuses on proving a property of a recursive sequence defined using floor and ceiling functions through strong mathematical induction. The sequence starts with c_0 = 1 and c_1 = 1, and for any integer k greater than or equal to 2, it is defined as c_k = c_(floor(k/2)) + c_(ceil(k/2)). The aim is to demonstrate that c_n equals n for each integer n greater than or equal to 1. The proof begins by verifying base cases where c_1 is 1 and c_2, calculated as c_1 + c_1, equals 2, both satisfying the sequence's rule. The inductive step assumes the property holds for all integers up to k, and extends it to k + 1. Using the recursive definition, c_(k+1) is expressed as c_(floor((k+1)/2)) + c_(ceil((k+1)/2)), and by applying the inductive hypothesis, these terms match their respective indices, thus proving c_(k+1) equals k + 1. The proof shows that the sequence's linear growth pattern, ensured by the recursive definition involving floor and ceiling operations, remains valid across all natural numbers.

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