Chapter 12.5, Problem 19E

To determine

To Prove: $8^n-3^n$ is divisible by 5 for all natural numbers $n$ .

Explanation of Solution

Given information:

Use Mathematical induction to prove the above mentioned formula.

Concept used:

1.By using Mathematical induction technique we need to prove that this formula holds True for all natural numbers. The first step is to prove True for $n=1$ .

2.Second step by using induction hypothesis of mathematical induction we assume for $n=k$ given formula is True.

3.Third step is we need to prove that above formula is True for $n=k+1$ by using formula is True for $n=k$ .

Proof:

First, we need to prove the above formula is True for $n=1$ .

Given formula is

  $8^n-3^n$ is divisible by 5

Put $n=1$

We get

  $\begin{array}{l}{8}^n-3^n=8^1-3^1\\ \Rightarrow 8^n-3^n=8-3\\ \Rightarrow 8^n-3^n=5\end{array}$

5 is divisible by 5.

It True for $n=1$ .

Let us assume it is True for $n=k$

So

  $8^k-3^k$ is divisible by 5 is True

We need to prove it is True for $n=k+1$

  $\begin{array}{l}{8}^n-3^n=8^{k+1}-3^{k+1}\\ \Rightarrow 8^n-3^n=8\left(8^k\right)-3\left(3^k\right)\\ \Rightarrow 8^n-3^n=\left(5+3\right)\left(8^k\right)-3\left(3^k\right)\\ \Rightarrow 8^n-3^n=5\left(8^k\right)+3\left(8^k\right)-3\left(3^k\right)\\ \Rightarrow 8^n-3^n=5\left(8^k\right)+3\left(8^k-3^k\right)\end{array}$

Here from our assumption $8^k-3^k$ is divisible by 5,

So $3\left(8^k-3^k\right)$ is also divisible by 5

and

  $5\left(8^k\right)$ is also divisible by 5 because it is multiple of 5.

So

  $8^n-3^n$ for $n=k+1$ is divisible by 5.

And hence the proof by Mathematical induction.