Chapter 10.7, Problem 17PPS

To determine

To prove that $5^n+3$ is divisible by 4 for all natural numbers n.

Explanation of Solution

Given:

  $5^n+3$

Concept Used:

An identity can be proved using mathematical induction in two steps:

1. First prove the identity is true for $n=1$ .
2. Prove that if it is true for $n=k$ then it is also true for $n=k+1$ .

Calculation:

In order to prove that $5^n+3$ is divisible by 4 for all natural numbers n, use mathematical induction.

First, let’s check if this is true for $n=1$ .

For $n=1$

  $\begin{array}{c}{5}^1+3=5+3\\ =8\end{array}$

And 8 is divisible by 4. Thus, the given identity is true for $n=1$ .

Now, let it is true for $n=k;\text{ for some }k>1$ .

Thus, $\left(5^k+3\right)$ is divisible by 4, i.e.

  $5^k+3=4m$ , for some integer m.

Now, let’s show that it is also true for $n=k+1$ , i.e. prove that $\left(5^{k+1}+3\right)$ is also divisible by 4. Since,

  $\begin{array}{c}{5}^{k+1}+3=5^k\cdot 5+3\\ =\left(4m-3\right)\cdot 5+3\left(\begin{array}{l}\because 5^k+3=4m\\ \Rightarrow 5^k=4m-3\end{array}\right)\\ =4m\cdot 5-15+3\\ =4m\cdot 5-12\\ =4\left(5m-3\right)\end{array}$

Thus, $\left(5^{k+1}+3\right)$ is a multiple of 4, i.e. divisible by 4, i.e. the identity is also true for $n=k+1$ , when it is true for $n=k$ .

Thus, by mathematical induction it is proved that $5^n+3$ is divisible by 4 for all natural numbers n.