Problem 28 - College Algebra with Modeling and Visualization

Question

Prove the statement by mathematical induction. \(4^{n}>n^{4}\) for \(n \geq 5\)

Step-by-Step Solution

Answer

By induction, \(4^n > n^4\) for all \(n \geq 5\).

Step 1: Base Case

First, verify the statement for the base case, \(n = 5\). Calculate \(4^5\) and \(5^4\). We find that \(4^5 = 1024\) and \(5^4 = 625\). Clearly, \(1024 > 625\). Thus, the base case is true.

Step 2: Inductive Hypothesis

Assume the statement is true for some \(n = k\), where \(k \geq 5\). This means we assume \(4^k > k^4\).

Step 3: Inductive Step

We need to prove that if the statement holds for \(n = k\), it also holds for \(n = k + 1\). This means showing \(4^{k+1} > (k+1)^4\).

Step 4: Express Inductive Step

Use the inductive hypothesis: \(4^{k+1} = 4 \cdot 4^k\). By the inductive hypothesis, we have \(4^k > k^4\). Thus, \(4^{k+1} = 4 \cdot 4^k > 4 \cdot k^4\).

Step 5: Comparing Inequalities

To complete the inductive step, show that \(4k^4 \geq (k+1)^4\) when \(k \geq 5\). Expand \((k+1)^4\) as \(k^4 + 4k^3 + 6k^2 + 4k + 1\). Notice that for \(k \geq 5\), the inequality holds because the terms involving \(k\) are dominantly large.

Step 6: Verification of Inductive Step

Test that for \(k=5\), the inequality holds. We have \(4 \times 5^4 = 2500\), and \((5+1)^4 = 6^4 = 1296\), thus \(2500 > 1296\). This confirms the truth of the inequality for \(k+1\).

Step 7: Concluding the Induction

Having verified the base case and established that \(4^k > k^4\) implies \(4^{k+1} > (k+1)^4\) for all \(k \geq 5\), by the principle of mathematical induction, \(4^n > n^4\) is true for all \(n \geq 5\).

Key Concepts

Base Case

In mathematical induction, the base case serves as the foundation of the proof process. It's like checking the first step on a staircase to ensure it's stable before you climb the whole flight. For our exercise, the statement we wish to prove is \(4^n > n^4\) for \(n \geq 5\).

To begin, we evaluate whether the statement holds when \(n = 5\). We compute the values:

• \(4^5 = 1024\)
• \(5^4 = 625\)

Since \(1024\) is greater than \(625\), the base case is true. This initial success means we can confidently move forward with our induction process, knowing our starting point is correct.

Inductive Hypothesis

The inductive hypothesis is an assumption we make during the process of mathematical induction. It's a crucial part of the proof where you suppose that a statement is true for some arbitrary case \(n = k\).

For our problem, we assume that \(4^k > k^4\) where \(k \geq 5\). This is like believing a pattern holds for a particular step on the staircase.

• Think of this as setting a condition that helps us prove the next step.

With this assumption in hand, our job is to show that if it holds for \(n = k\), it will also hold for \(n = k + 1\). This leap from \(k\) to \(k + 1\) is what confirms the pattern persists throughout.

Inductive Step

The inductive step is where the magic of induction truly takes place. Here, you prove that if a statement holds for \(n = k\), it must also hold for \(n = k + 1\).

For this exercise, our task is to demonstrate \(4^{k+1} > (k+1)^4\) using the inductive hypothesis \(4^k > k^4\). Here's how we approach it:

• Write \(4^{k+1}\) as \(4 \cdot 4^k\).
• Since \(4^k > k^4\), substitute to get \(4 \cdot 4^k > 4 \cdot k^4\).

The breakthrough is simplifying \((k+1)^4\) and showing it's less than \(4 \cdot k^4\) for \(k \geq 5\). This step verifies not only that the pattern holds for an arbitrary \(k\), but also extends naturally to \(k + 1\). Essentially, this is how you walk up the entire staircase after confirming your base step.

Inequality Proof

Inequality proofs in induction can be tricky, often involving strategic comparisons of terms.

To prove \(4k^4 \geq (k+1)^4\) for \(k \geq 5\), we expand \((k+1)^4\) into simpler terms to see the whole picture:

• \(k^4 + 4k^3 + 6k^2 + 4k + 1\)

Breaking it down lets us assess how significant each term is when \(k\) is relatively large, like 5 or more. The terms involving \(k\) are substantial, which makes the inequality hold true.

For verification, we compute for \(k = 5\), where:

• \(4 \times 5^4 = 2500\)
• \(6^4 = 1296\)

Since \(2500 > 1296\), the inequality stands firm. This logic and evaluation authenticate our inductive step and complete the induction, confirming the truth of \(4^n > n^4\) for all \(n \geq 5\).