Problem 3 - A Graphical Approach to Precalculus with Limits

Question

CONCEPT CHECK For which positive integers is the statement \(2^{n}>2 n\) not true?

Step-by-Step Solution

Answer

The statement is not true for \(n = 1\) and \(n = 2\).

Step 1: Understand the Inequality

The expression we need to evaluate is \(2^n > 2n\). We need to find for which positive integers \(n\) this inequality does not hold. This means we need to determine when \(2^n \leq 2n\).

Step 2: Test Small Values of n

Let's start by testing small values of \(n\).- For \(n = 1\): \(2^1 = 2\) and \(2 \times 1 = 2\). Thus, \(2^1 \leq 2 \times 1\) is true.- For \(n = 2\): \(2^2 = 4\) and \(2 \times 2 = 4\). Thus, \(2^2 \leq 2 \times 2\) is also true.For \(n = 1\) and \(n = 2\), the inequality \(2^n > 2n\) is not true.

Step 3: Check if the Pattern Holds

Now, let's see if the pattern holds for larger values of \(n\).- For \(n = 3\): \(2^3 = 8\) and \(2 \times 3 = 6\). Thus, \(2^3 > 2 \times 3\), the inequality holds.- For larger values of \(n\), observe that \(2^n\) grows exponentially while \(2n\) grows linearly. Therefore, for \(n > 2\), \(2^n\) is always greater than \(2n\).

Step 4: Conclusion

The statement \(2^n > 2n\) is not true for \(n = 1\) and \(n = 2\). For any \(n > 2\), the inequality does hold true. Therefore, the positive integers for which the statement is not true are \(n = 1\) and \(n = 2\).

Key Concepts

Integer Evaluation

In mathematics, integers are whole numbers that can be positive, negative, or zero. In this exercise, we aim to determine which positive integers make the inequality \(2^n > 2n\) false. Evaluating inequalities like this involves testing specific integer values to see where the inequality holds or does not hold.

By starting with simple, small integers, we simplify the problem considerably. Testing these can give us a clear picture of the inequality's behavior. For example:

• When \(n = 1\), we calculate \(2^1 = 2\) and \(2 \times 1 = 2\), so \(2^1 \leq 2 \times 1\). Thus, the inequality is not true for \(n = 1\).
• Repeating this for \(n = 2\), where \(2^2 = 4\) and \(2 \times 2 = 4\), we also find \(2^2 \leq 2 \times 2\). Hence, the inequality does not hold for \(n = 2\).

In this way, evaluating the inequality for integers helps us identify the exact boundary where it transitions from false to true.

Exponential Growth

Exponential growth refers to the process of increasing in number by a constant multiplicative factor over successive periods. The function \(2^n\) in our inequality represents exponential growth.

As \(n\) gets larger, \(2^n\) increases rapidly. For instance:

• For \(n = 3\), we find \(2^3 = 8\), while \(2 \times 3 = 6\). Here, \(2^3\) is greater, validating the inequality.
• Beyond this, for every integer greater than 2, \(2^n\) outpaces \(2n\) due to its exponential nature. This means the inequality \(2^n > 2n\) holds true for all \(n > 2\).

Understanding exponential growth helps us see why the inequality holds true past a certain point. It illustrates the powerful compounding effect inherent in exponential functions, making them grow much faster than linear ones.

Linear Growth

Linear growth is characterized by a consistent, even increase. In our inequality, the term \(2n\) exhibits linear growth. For each unit increase in \(n\), two units are added to \(2n\). This simplicity helps clarify why the inequality changes at higher values of \(n\).

Unlike exponential growth, linear growth keeps pace steadily without abrupt increases. This is why:

• Initially, at smaller values like \(n = 1\) and \(n = 2\), linear increases can match or exceed small exponential values, making \(2^n \leq 2n\) true.
• However, as \(n\) grows larger, the steady pace of linear growth is simply outstripped by the accelerating increase of exponential growth, reversing the inequality's truth state.

By contrasting linear with exponential growth, we can understand how the inequality \(2^n > 2n\) becomes permanently true beyond a certain threshold. The consistent nature of linear growth provides a steady baseline, against which the swiftly increasing pace of exponential growth stands out.