Inductive step in mathematical induction – UGC NET Question

In mathematical induction, there are two main parts: the base case and the inductive step. The inductive step assumes that the statement holds for n = k (the induction hypothesis) and then proves that it must also hold for n = k + 1. This step is called the inductive step because it extends the truth from one case to the next. Hence, the description in the question refers to the inductive step.

Option A:
The basis or base step is where we verify that the statement holds for the initial value, usually n = 1 or n = 0. It does not involve the assumption "if true for k then true for k + 1." Therefore, basis does not match the part described in the question.

Option B:
Verification is a general term that could refer to checking any part of the proof. It is not the standard technical name for the specific step that connects n = k to n = k + 1. Hence, it is too vague and not suitable as an answer.

Option C:
The inductive step is correct because it formalises the logical link needed to propagate the truth of the statement from one natural number to the next. Once both the base case and inductive step are established, the statement is proved for all natural numbers in the domain. Recognising this terminology is important for understanding proofs in discrete mathematics and reasoning.

Option D:
Concluding step might describe the final remarks of a proof, but it is not a standard name for the process of moving from k to k + 1. The main logical engine in induction is the inductive step, not a generic conclusion. Thus, this option is not accurate.