Problem 29 - Contemporary Precalculus

Question

Use the Extended Principle of Mathematical Induction (Exercise 28 ) to prove the given statement. \(2 n-4>n\) for every \(n \geq 5 .\) (Use 5 for \(q\) here.)

Step-by-Step Solution

Answer

Question: Prove the inequality \(2n - 4 > n\) is true for every \(n \geq 5\) using the Extended Principle of Mathematical Induction.

Step 1: Base Case

When \(n = 5\), let's check if the inequality \(2n - 4 > n\) holds: $2(5) - 4 > 5 10 - 4 > 5 6 > 5$ The inequality is true for the base case when \(n = 5\).

Step 2: Inductive Hypothesis

Assume that the inequality is true for \(n = k\), where \(k \geq 5\): \(2k - 4 > k\)

Step 3: Inductive Step

Now we need to prove that if the inequality is true for \(n = k\), then it is also true for \(n = k + 1\). Let's substitute \(n = k + 1\) in the inequality: $2(k + 1) - 4 > k + 1 2k + 2 - 4 > k + 1 2k - 2 > k + 1$ To prove this inequality using the inductive hypothesis, subtract \(k\) from both sides: \(k - 2 > 1\) Since \(k \geq 5\), this inequality is true: $5 - 2 > 1 3 > 1$ Thus, the inequality is true for \(n = k + 1\) as well. By using the Extended Principle of Mathematical Induction, we have proved that the inequality \(2n - 4 > n\) is true for every \(n \geq 5\).

Key Concepts

Base Case

When working with mathematical induction, one begins by verifying the base case. Think of the base case as the first step or foundation of a domino effect in a line of dominos. If the first domino successfully falls, it may trigger the next, and so forth.

In this specific problem, the statement we aim to prove is that the inequality \(2n - 4 > n\) holds for every \(n \geq 5\). We start by checking the smallest value of \(n\), which is 5 in this context. Plugging in \(n = 5\), the inequality becomes:

• \(2(5) - 4 > 5\)
• \(10 - 4 > 5\)
• \(6 > 5\)

This result confirms that the base case holds true. Think of it as confirming that the first domino (`\(n = 5\)`) has indeed tipped over, triggering the sequence to start. Without this crucial step, we'd have no starting point for proving the inequality on a broader scale.

Inductive Hypothesis

The inductive hypothesis is a powerful tool used to assume the statement or inequality we're trying to prove holds true for a certain case \(n = k\).

In simpler terms, one temporarily assumes the statement is true for this arbitrary integer \(k\) (where \(k \geq 5\) for our problem). So here, we assume \(2k - 4 > k\).
This is a crucial assumption because it sets up the step to show that if the statement is true for one arbitrary case, it should also be true for the next case, \(n = k + 1\).

With the inductive hypothesis, we focus on a potential pattern that connects consecutive numbers. Think of it as a claim about the whole row of dominos tipping over, provided that one particular domino falls.This hypothesis is what ties the base case to the more complex inductive step, forming a continuous link across all cases starting from \(n = 5\). It’s a stepping stone to move from the known to the unknown.

Inductive Step

This step completes the induction process by demonstrating that if the hypothesis is valid for \(n = k\), it must also be valid for \(n = k + 1\), thereby proving the statement for all \(n\geq 5\).

For the inductive step, we replace \(n\) in the inequality with \(k + 1\) to get:

• \(2(k + 1) - 4 > k + 1\)
• Expanding gives \(2k + 2 - 4 > k + 1\)
• Simplifying leads to \(2k - 2 > k + 1\)

Using the inductive hypothesis \(2k - 4 > k\), subtract \(k\) from both sides:

• \(k - 2 > 1\)

Since we start from \(k \geq 5\), it's evident that:

• \(5 - 2 > 1\)
• \(3 > 1\), which is true.

Because the inequality holds true for \(n = k + 1\), the domino continues to fall, completing the logic chain.

By connecting the base case and adding more dominos through our inductive step, we’ve managed to prove the entire sequence of dominos will fall, or in mathematical terms, that the inequality holds for all applicable values of \(n\). This elegant process illustrates induction's beauty in weaving simple truths into more complex, universal statements.