Question

In Problem, prove each statement.

If $x>1$then $x^n>1$

Short answer

The given statement is shown.

Step-by-step solution

Step 1. Given information

$x>1$

Step 2. Prove that xn>1.

Put $n=1$in $x^n>1$to check if the statement is true for $n=1$

width="68">x1=x>1

Thus, the statement is true for $n=1$

Assume that the statement is true for $n=k$

Thus, $x^k>1$.

Find whether $x^{k+1}>1$.

Multiply both the sides of $x^k>1$by width="10">x

$x^k·x>x$

width="61">xk+1>x

$x^{k+1}>1$because $x>1$

Thus, the statement is true for$k+1$

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement is true for all natural numbers.