In an induction proof, what is the purpose of the base case?

In an induction proof, what is the purpose of the base case?

• A. To prove the statement for n = k
• B. To verify the statement for the initial value
• C. To prove the general case directly
• D. To derive a contradiction

Answer: (B)

The base case anchors the whole induction. It verifies the statement for the initial value of the natural numbers (often n = 0 or n = 1), giving a concrete starting point from which the inductive step can build. Once that starting truth is established, you prove that if the statement holds for one value n, it must hold for the next value n+1. This creates a chain that, by the principle of mathematical induction, shows the statement is true for all n in the domain. If the base case isn’t true, there’s no foundation for the inductive step to carry forward. For example, checking n = 1 in a sum formula confirms the base instance, and then the inductive step extends that truth to all larger n.

The base case anchors the whole induction. It verifies the statement for the initial value of the natural numbers (often n = 0 or n = 1), giving a concrete starting point from which the inductive step can build. Once that starting truth is established, you prove that if the statement holds for one value n, it must hold for the next value n+1. This creates a chain that, by the principle of mathematical induction, shows the statement is true for all n in the domain. If the base case isn’t true, there’s no foundation for the inductive step to carry forward. For example, checking n = 1 in a sum formula confirms the base instance, and then the inductive step extends that truth to all larger n.