Mathematical induction stands as a cornerstone proof technique in discrete mathematics, enabling the verification of propositions defined over the set of natural numbers. This review offers a comprehensive analysis of the principle of mathematical induction, elucidating its theoretical foundations and its extensive, practical applications across diverse fields such as computer science, cryptography, computational finance, and systems biology. The article delves into specific use cases, including the analysis of algorithmic complexity, the validation of cryptographic protocols like RSA, the modeling of compound interest in finance, and the study of population dynamics. Through detailed case studies and proofs, we demonstrate that inductive reasoning is not merely an abstract mathematical tool but a vital methodology for solving recursive problems in both theoretical and applied disciplines.