Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What does mathematical induction typically require to prove a statement?

Showing it is true for all integers
Proving the base case and the inductive step
Using direct proof for each case
Formulating a counterexample

The correct answer is: Proving the base case and the inductive step

Mathematical induction is a standard technique used to prove that a statement is true for all natural numbers, and it requires two critical components: the base case and the inductive step. The base case involves demonstrating that the statement holds true for the initial value, often when n = 1. This step is essential because it establishes the foundation for the induction process. Once the base case is confirmed, the next step is the inductive step, where we assume the statement is true for a given integer k and then prove that it must also be true for the next integer, k + 1. This logical progression is what allows us to conclude that the statement holds for all natural numbers. This method is distinctive because, unlike proving a statement true for all integers outright, induction provides a systematic approach based on verifying an initial case and then extending that truth to all other cases. Using direct proof for each case, as suggested in one of the options, would be impractical for statements involving infinitely many integers. Lastly, formulating a counterexample is not relevant to induction, as it is used to disprove statements rather than prove them.