Problem 16 - Precalculus Mathematics for Calculus

Question

Show that \(5^{n}-1\) is divisible by 4 for all natural numbers \(n\)

Step-by-Step Solution

Answer

By induction, \(5^n - 1\) is divisible by 4 for all natural numbers \(n\).

Step 1: Base Case

First, we check the base case for the smallest natural number, which is 1. Substitute \(n = 1\) into the expression \(5^n - 1\):\[5^1 - 1 = 5 - 1 = 4\]Since 4 is divisible by 4, the base case holds true.

Step 2: Inductive Hypothesis

Assume that the statement is true for some natural number \(k\), i.e., assume that \(5^k - 1\) is divisible by 4. This means that there is an integer \(m\) such that:\[5^k - 1 = 4m\]

Step 3: Inductive Step

We need to show that if the statement holds for \(n = k\), then it must also hold for \(n = k + 1\). Consider:\[5^{k+1} - 1 = 5 \cdot 5^k - 1 = 5 \cdot 5^k - 5 + 4\]Rearrange and substitute using the inductive hypothesis:\[= 5(5^k - 1) + 4 = 5(4m) + 4 = 20m + 4\]Thus, \(5^{k+1} - 1 = 4(5m + 1)\). This expression is clearly divisible by 4.

Step 4: Conclusion

Since the base case is true and the inductive step holds, by mathematical induction, \(5^n - 1\) is divisible by 4 for all natural numbers \(n\).

Key Concepts

Divisibility

Divisibility is a fundamental concept in mathematics where one integer can be divided by another without leaving a remainder. In simpler terms, given two integers, A and B, if A can be divided by B and the result is a whole number, then A is said to be divisible by B. In the context of our problem, divisibility is about confirming that \(5^n - 1\) can always be divided by 4 cleanly, regardless of what natural number \(n\) is.

Here's why divisibility is important:

• It helps in simplifying equations.
• It plays a key role in number theory and encryption.
• It's crucial for solving problems related to fractions and ratios.

In this exercise, to say \(5^n - 1\) is divisible by 4 means whenever you calculate \(5^n - 1\), if you divide by 4, you will always get a whole number.

Base Case

The base case is the foundation of mathematical induction, acting as a starting point for proving statements about natural numbers. The idea is to show that a hypothesis is true for the initial value, often \(n = 1\) or sometimes \(n = 0\). If this first step doesn't work, the entire inductive process falls apart.In this problem, the base case involves checking whether \(5^1 - 1\) is divisible by 4. Calculating, we find:\[5^1 - 1 = 5 - 1 = 4\]Since 4 is divisible by 4, our base case holds true.The base case is crucial because:

• It confirms the initial truth of the statement.
• It sets the condition for further steps in induction.
• If the base case fails, the proof using induction cannot proceed.

By establishing that \(5^1 - 1\) is divisible by 4, we set the stage for proving it for all other natural numbers.

Inductive Hypothesis

The inductive hypothesis, or the assumption step, is where we assume that our statement is true for some arbitrary natural number \(k\). This step is like a set-up: we don't prove anything outright yet, but we prepare to move towards a general proof.Here, we assume that \(5^k - 1\) is divisible by 4, meaning there exists an integer \(m\) such that:\[5^k - 1 = 4m\]This assumption is pivotal because:

• It allows us to use existing truth to extend proof.
• Sets a specific condition that the next step builds on.
• Bridges the gap between specific and general cases.

Once the inductive hypothesis is secure, we can look towards the next steps: using it to go from \(n = k\) to \(n = k + 1\).

Inductive Step

The inductive step is where we take our hypothesis for \(n = k\) and use it to prove the case for \(n = k + 1\). This transition is the most important, as it convinces us that if the statement holds for one number, it holds for the next, thereby for all natural numbers.Here, starting from:\[5^{k+1} - 1 = 5 \cdot 5^k - 1\]We aim to show it is divisible by 4. Using the inductive hypothesis \(5^k - 1 = 4m\), the expression becomes:\[5 \cdot 5^k - 5 + 4 = 5(4m) + 4 = 20m + 4\]Rearranging shows:\[5^{k+1} - 1 = 4(5m + 1)\]Here, it's evident the expression is clearly divisible by 4.The inductive step works because:

• It leverages the assumption from the hypothesis step.
• Ensures the truth extends to all further cases.
• Solidifies the conclusion of the proof process.

By successfully completing the inductive step, we validate our initial proposition for every natural number, thanks to the framework of mathematical induction.