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This is a practice question from my Data Structures course:
h(n) = {1 if n = 1; 3 ×ばつ h(n - 1) - 1 otherwise}
Rewrite h(n) using an explicit formula that is not recursive (i.e., succinctly express h(n) as a polynomial/exponential function of n). Also, prove the correctness of your expression using mathematical induction.
Please explain how to do this (as opposed to simply providing the answer).
Thank you for your assistance.
In the realm of mathematical recursion, the quest to find explicit formulas for recursive functions is a fascinating journey. In this context, encountered the recursive function h(n), defined as h(n) = 1 if n = 1, and h(n) = 3 * h(n - 1) - 1 otherwise. The challenge at hand is to transform this recursive definition into an explicit formula—a succinct mathematical expression that directly relates h(n) to n. Through careful observation and pattern recognition, embark to unveil the hidden relationship between h(n) and n. Find the explicit formula but also rigorously prove its correctness using mathematical induction, solidifying our understanding of h(n) as a polynomial/exponential function of n.
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