Chapter 9.4, Problem 77E

To determine

Determine whether the given statement is true or false.

Answer to Problem 77E

The statement is true.

Explanation of Solution

Given:

If $P_k$ is true and $P_k$ implies $P_{k+1}$ , it is possible only if $P_1$ is true, from the principal of mathematical induction.

The statement is true.

As it is given $P_k$ is true and $P_k$ implies $P_{k+1}$ , it is possible only if $P_1$ is true, from the principal of mathematical induction.

The Principle of Mathematical Induction:

Let $P_n$ be a statement involving the positive integer n.

If,

1. $P_1$ is true, and

2. Fir every positive integer k, the truth of k implies the truth of $P_{k+1}$ then the statement $P_n$ must be true for all positive integers $n$ .