Chapter 11.1, Problem 37SR

To determine

Prove that the statement is true for all natural numbers.

Answer to Problem 37SR

Proved .

Explanation of Solution

Given information:

  $9^n-1$ is divisible by $8$ .

Calculation:

Consider the statement $9^n-1$ is divisible by $8$ is true,

  $\begin{array}{l}put,n=1\\ 9^1-1=9-1\\ 9^1-1=8\end{array}$

The statement is true for $n=1$ .

Now assume $9^k-1$ is divisible by $8$ is true for some natural numbers $k$ for some value of $m$ .

  $\begin{array}{l}{9}^k-1+1=8m+1\\ \text{Multiply bith side by }9,\\ 9^k\times 9=\left(8m+1\right)\times 9\\ 9^{k+1}=72m+9\\ 9^{k+1}-1=72m+9-1\\ =72m+8\\ =8\left(9m+1\right)\end{array}$

Thus $m$ is a natural number, $9m+1$ is a natural number, and $8\left(9m+1\right)$ is divided by $8$ .

Hence, proved $9^n-1$ is divisible by $8$ is true for all natural numbers.