Problem 16 Using induction, verify the ineq... [FREE SOLUTION]

Chapter 2: Problem 16

Using induction, verify the inequality. $$ 2^{n} \geq n^{2}, n=4,5, \ldots $$

Short Answer

Expert verified

By induction, we can confirm the inequality $2^{n} \geq n^{2}$ holds for all $n=4,5,\ldots$. First, we demonstrate the base case $n=4$, resulting in $2^{4}=16$ which equals $4^{2}$. Next, we assumed that for an arbitrary $2^{n} \geq n^{2}$, proving that $2^{n+1} \geq (n+1)^{2}$ confirmed the inequality for $n+1$. Hence by mathematical induction, the inequality $2^{n} \geq n^{2}$ is true for all $n=4,5,\ldots$.

Step by step solution

Base Case

First, let's validate the base case. The base case for this problem is $n=4$, as the inequality is stated to be true for $n=4,5,\ldots$. We substitute $n=4$ into the inequality $2^{n} \geq n^{2}$ to see if it holds true. Upon substitution we find that $2^{4}=16$ and $4^{2}=16$. Since both are equal (and thus $2^{4}$ is not less than $4^{2}$), the inequality is correct for our base case $n=4$.

Inductive Step

The aim of the inductive step is to verify if the inequality holds for $n+1$ given it is true for $n$.\nWe first assume that the inequality $2^{n} \geq n^{2}$ is true.\nWe need to prove: $2^{n+1} \geq (n+1)^{2}$.\nThen, we express $2^{n+1}$ as $2 \cdot 2^{n}$ and try to demonstrate whether it is larger or equal to $(n+1)^{2}$.\nGiven $2^{n} \geq n^{2}$ (our induction hypothesis), and, by assuming $n>=4$, we can express $2^{n}$ as higher or equal to $2(n+1)$.\nIf so, then $2 \cdot 2^{n} \geq 2(n+1)$.\nSimplifying gives us $2^{n+1} \geq 2n+2 = n^{2}+2n+1+1 = (n+1)^{2}+1$.\nThis confirms the inequality for $n+1$.

Conclusion

Having established that the inequality holds for the base case and that, if it's true for an arbitrary instance $n$, it's also true for $n+1$, our mathematical induction is complete.\nTherefore, by the principle of mathematical induction, the inequality $2^{n} \geq n^{2}$ is valid for all $n=4,5,\ldots$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Proof

Proving inequalities is essential for understanding how quantities relate to each other under certain conditions. In the context of mathematical induction, an inequality proof establishes that a given relationship holds true across an infinite sequence of numbers. In our exercise, we examine the inequality $2^{n} \geq n^{2}$, for all integers $n$ starting from 4.

To conduct an inequality proof using mathematical induction, we start by showing that the inequality is valid for a specific case鈥攌nown as the base case. We then assume it's true for an arbitrary case $n$, and use this assumption to demonstrate it remains true for the next case, $n+1$. This progression from known to unknown instances is what allows us to extend the proof to all numbers in the sequence. Explaining this concept in simple, relatable terms enhances comprehension, ensuring students can apply the principle to similar problems.

Inductive Step

The inductive step is the heart of the induction process. After verifying the base case, we hypothesize that our inequality holds for a certain $n$, paving the way to show its truth for $n+1$.

For our example, we take the presumed truth that $2^{n} \geq n^{2}$ for some $n$ and work to show $2^{n+1} \geq (n+1)^{2}$. We do this by manipulating the expression, connecting $2^{n+1}$ to our hypothesis, and proving it exceeds or equals $ (n+1)^{2}$.

This pivotal step requires meticulous attention to algebraic rearrangement and logical reasoning. To make it digestible, it's worthwhile to step through the algebra slowly, ensuring that each transition is clear and justified. This clarity aids students in developing strong conceptual understanding and analytical skills.

Base Case

The base case serves as the starting point for any proof by induction. It demonstrates that the property you're proving is true for the first item in your series. In our inequality $2^{n} \geq n^{2}$, the base case is $n=4$, the smallest integer we're considering.

Validating the base case involves simply plugging $n=4$ into the inequality and checking whether the statement holds. If the base case doesn't hold, the entire inductive argument falls apart. We find that $2^{4}=16$ and $4^{2}=16$, so indeed, $2^{n} \geq n^{2}$ is true for $n=4$. It's crucial to explain the importance of the base case to students, as it is often overlooked but is the foundation upon which the inductive steps are built. Simple analogies, such as the base of a ladder from which one climbs rung by rung, can be very effective in conveying this concept.

91影视

Using induction, verify the inequality. $$ 2^{n} \geq n^{2}, n=4,5, \ldots $$

Short Answer

Step by step solution

Base Case

Inductive Step

Conclusion

Key Concepts

Inequality Proof

Inductive Step

Base Case

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