Problem 32 - A Graphical Approach to College Algebra

Question

Prove each statement by mathematical induction. $$4^{n}>n^{4}, \text { for } n \geq 5$$

Step-by-Step Solution

Answer

The inequality holds for all \( n \geq 5 \) by induction.

Step 1: Base Case

To start the proof by induction, we check if the statement is true for the base case of the smallest value, which is \( n = 5 \). Let's calculate \( 4^5 \) and \( 5^4 \):- \( 4^5 = 1024 \)- \( 5^4 = 625 \)Since \( 1024 > 625 \), the statement holds for \( n = 5 \). Thus, the base case is true.

Step 2: Inductive Hypothesis

Assume that the given inequality \( 4^n > n^4 \) is true for some arbitrary integer \( k \geq 5 \). This means we assume:\[ 4^k > k^4 \]

Step 3: Inductive Step

Now, we need to prove that if the statement holds for \( n = k \), then it also holds for \( n = k + 1 \). So, we must show:\[ 4^{k+1} > (k+1)^4 \]

Step 4: Express \(4^{k+1}\) Using the Inductive Hypothesis

The expression for \( 4^{k+1} \) in terms of \( 4^k \) is:\[ 4^{k+1} = 4 \cdot 4^k \]Using the inductive hypothesis, where \( 4^k > k^4 \), we can substitute:\[ 4^{k+1} = 4 \cdot 4^k > 4 \cdot k^4 \]

Step 5: Show \(4 \cdot k^4 > (k+1)^4\) for \(k \geq 5\)

Simplify the inequality to:\[ 4 \cdot k^4 > (k+1)^4 \]Expanding \((k+1)^4\), we get:\[(k+1)^4 = k^4 + 4k^3 + 6k^2 + 4k + 1\]Compare the expanded polynomial:\[ 4k^4 > k^4 + 4k^3 + 6k^2 + 4k + 1 \]This can be shown to hold true for \(k \geq 5\) by comparing the dominant term \(4k^4\) with the remainder of the polynomial.

Step 6: Verify the Inductive Step

For \(k = 5\):- Calculate both sides: - Left side: \(4 \times 5^4 = 2500\) - Right side: \(6 \times 5^3 = 750, 6 \times 5^2 = 150, 4 \times 5 = 20\) gives; \(5^4 + 4 \times 5^3 + 6 \times 5^2 + 4 \times 5 + 1 = 1296\)Thus, \( 2500 > 1296 \), showing the inequality holds.For the remaining terms, analyze behavior as \(k\) increases by considering ratios or inequalities.

Step 7: Conclusion

Since the base case is true and the inductive step holds, we have proved by induction that \( 4^n > n^4 \) is true for all \( n \geq 5 \).

Key Concepts

Inequalities

In mathematics, inequalities express the relationship of being greater or less between two values. When we say that \(4^n > n^4\), we are describing an inequality. This particular inequality is saying that the value produced when 4 is raised to the power of \(n\) is greater than the value produced when \(n\) is raised to the power of 4, for \(n \geq 5\).

When solving inequalities, it is important to compare the expressions on each side to determine when one is greater than the other. In this exercise, we use mathematical induction to prove the inequality holds for all values greater than or equal to 5. This involves validating it at a starting point (base case) and proving it for all successive numbers (inductive step).

Base Case

The base case is the first step in mathematical induction and serves to establish that the statement is true for an initial value of \(n\). For this problem, we start with \(n = 5\).

Calculating the left side of the inequality, we have \(4^5 = 1024\). For the right side, \(5^4 = 625\). Clearly, \(1024 > 625\), proving the base case as true. Without verifying the base case, the induction process cannot proceed, because it forms the foundation upon which the rest of the proof is built.

Inductive Step

After confirming the base case, the next stage is the inductive step. This step involves assuming the inequality is true for an arbitrary number \(k\), and proving it holds for \(k + 1\).

The assumption, known as the inductive hypothesis, is \(4^k > k^4\). The goal is to use this to show \(4^{k+1} > (k+1)^4\).

The expression \(4^{k+1}\) can be rewritten as \(4 \cdot 4^k\). Using the inductive hypothesis, we substitute to get \(4 \cdot k^4\), then proceed to show that this is greater than \((k+1)^4\).

Breaking down the polynomial \((k+1)^4\) reveals additional terms like \(4k^3\), \(6k^2\), and others. The strategy involves comparing these terms collectively against \(4k^4\) to confirm that indeed the inequality holds. Successful completion of this step allows the induction process to validate the inequality for all integers greater than or equal to 5.

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a sum of terms raised to various powers. For the inductive proof of \(4^n > n^4\), we need to expand \((k+1)^4\).

The expansion of \((k+1)^4\) is calculated as follows:

• \((k+1)^4 = k^4 + 4k^3 + 6k^2 + 4k + 1\)

This fully expanded form breaks down the power into manageable parts for comparison with \(4k^4\).

In the induction step, each component of the polynomial is compared collectively against \(4k^4\). Because \(k\) is large (at least 5), the dominating term \(k^4\) is more impactful than the other smaller-degree terms. This insight helps to verify that \(4k^4\) will continue to exceed these polynomial terms for all allowed values of \(k\).