Which of the following illustrates the inductive step to prove a statement P(n

Discussion Forum : Principle Of Mathematical Induction

Question -

Which of the following illustrates the inductive step to prove a statement P(n) about natural numbers n by mathematical induction, where k is an arbitrary natural number?

Options:

A . P(1) is true

B . P(k) is true

C . P(k) is true

\Rightarrow

P(k+1) is true

D . none of these

Answer: Option C
:
C

Suppose there is a given statement P(n) involving natural numbers n such that

i) The statement is true for a specific natural number m i.e. P(m) is true. This is known as the base case.

ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n=k+1, i.e., the truth of P(k) implies the truth of P(k+1).
The second step is called the inductive step.

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