Chapter 13.3, Problem 53SR

To determine

To proof: The each statement is true for all the positive integers.

Explanation of Solution

Given:

The statement is given as:

  $4^n-1\text{is divisible by 3}$

Now, putting the value of the n equal to 1:

Hence,

  $\begin{array}{c}{4}^n-1=4^1-1\\ =4-1\\ =3\end{array}$

The statement is true for n equal to 1.

Assuming the given statement is true for $n=k\ge 1$ .

  $4^k-1$

As it is needed to obtain that it is divisible by 3. So, prove that the given statement is true for $n=k+1$ .

Now, proving $4^{k+1}-1\text{is divisible by 3}$ .

Now,

  $\begin{array}{c}{4}^{k+1}-1={4.4}^k-1\\ =\left(3+1\right)4^k-1\\ ={3.4}^k+4^k-1\end{array}$

As it is confirm that the ${3.4}^k$ is divisible by 3. By assuming $4^k-1$ is divisible by 3, hence, it is verified that $4^{k+1}-1$ is divisible by 3.

Therefore, the given statement is proved.