Chapter 12.4, Problem 23AYU

To determine

To prove: The given statement $x > 1$, then $x^n > 1$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

Answer to Problem 23AYU

As the statement is true for the natural number $\left(k + 1\right)$ terms, it is true for all natural numbers by the theorem of mathematical induction.

Explanation of Solution

Given:

Statements says the series $x > 1$, then $x^n > 1$ is true for all natural number.

Formula used:

The Principle of Mathematical Induction

Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:

CONDITION I: The statement is true for the natural number 1.

CONDITION II: If the statement is true for some natural number $k$, it is also true for the next natural number $k + 1$. Then the statement is true for all natural numbers.

Proof:

Consider the statement

$x > 1$, then $x^n > 1$   -----(1)

Step 1: Show that statement (1) is true for the natural number $n = 1$.

That is $x^1 = x > 1$ which is obvious from the statement. Hence the statement is true for the natural number $n = 1$.

Step 2: Assume that the statement is true for some natural number $k$.

That is $x > 1$, then $x^k > 1$   -----(2)

Step 3: Prove that the statement is true for the next natural number $k + 1$.

That is, to prove that $x > 1$, then $x^{k + 1} > 1$

Consider $x^{k + 1} = x^k. x^1$

In here, $x^k > 1$ and $x > 1$. The two numbers which are greater than 1, then their product must be greater than 1.

Hence $x^{k + 1} > 1$.

As the statement is true for the natural number $\left(k + 1\right)$ terms, it is true for all natural numbers by the theorem of mathematical induction.