Chapter 13.3, Problem 54SR

To determine

To prove: The given statement is true for all the positive integers.

Explanation of Solution

Given:

The statement is given as:

  $5^n+3\text{is divisible by 4}$

Put the value of the n equal to 1:

  $5^1+3=8$

Since, 8 is divisible by 4 then the statement is true for n equal to 1.

Now assuming for $n=k\ge 1$ , the statement is true.

As $5^k+3$ is divisible by 4. Hence, proving that the given statement is true for $n=k+1$ .

Here, it is required to show $5^{k+1}+3$ is divisible by 4.

  $\begin{array}{c}{5}^{k+1}+3={5.5}^k+3\\ =\left(4+1\right)5^k+3\\ ={4.5}^k+5^k+3\end{array}$

From the above expression, ${4.5}^k$ is divisible by 4. By the assumption $5^k+3$ will also be divisible by 4.

Therefore, it is simplified that the $5^{k+1}+3$ is divisible by 4.

Thus, the given statement is proved.