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?

• 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 1

The correct option is C

P(k) is true $\Rightarrow$ P(k+1) is true

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.