Chapter 9.5, Problem 41E

Solution

To identify : The correct option for the statement “ which of the following could be used to prove that for all positive integers $n$ ”

I. Mathematical induction

II. The formula for the sum of a finite arithmetic sequence

III. The formula for the sum of a finite geometric sequence

Option (A) I only

Option (B) I and II only

Option (C) I and III only

Option (D) II and III only

Option (E) I, II and III

option (A) is correct option for the statement “ which of the following could be used to prove that for all positive integers $n$ ”

I. Mathematical induction

II. The formula for the sum of a finite arithmetic sequence

III. The formula for the sum of a finite geometric sequence

Given information :

The statement for mathematical induction

Explanation :

Consider any statement “which of the following could be used to prove that for all positive integers $n$ ”

I. Mathematical induction

II. The formula for the sum of a finite arithmetic sequence

III. The formula for the sum of a finite geometric sequence

To prove any statement holds true for all positive integer $n$ , the mathematical induction is used.

To prove that it is true by mathematical induction, the following steps are followed.

Anchor step: Prove that any statement $P_n$ is true for $n=1$ .

Inductive hypothesis: Assume $P_n$ is true for $n=k$ .

Inductive Step: Prove that $P_n$ is true for $n=k+1$ .

Thus, the first step is anchor step where it is proved that $P_1$ is true.

So, the option (A) is correct.