Problem 21 - A Graphical Approach to College Algebra

Question

Find all positive integers n for which the given statement is not true. $$2^{n}>n^{2}$$

Step-by-Step Solution

Answer

The inequality \( 2^n > n^2 \) is false for \( n = 2, 3, \) and \( 4 \).

Step 1: Identify the Exercise Statement

The exercise asks us to find positive integer values of \( n \) for which the inequality \( 2^n > n^2 \) is not true. This means we need to find \( n \) such that \( 2^n \leq n^2 \).

Step 2: Test Small Positive Integers

Begin testing small positive integers to see where the inequality \( 2^n > n^2 \) breaks. Start with \( n = 1 \): - \( 2^1 = 2 \) and \( 1^2 = 1 \), so \( 2 > 1 \). Test \( n = 2 \):- \( 2^2 = 4 \) and \( 2^2 = 4 \), so \( 4 = 4 \). This is a case where the inequality does not hold (\( 2^n = n^2 \) instead).Now, test \( n = 3 \):- \( 2^3 = 8 \) and \( 3^2 = 9 \), so \( 8 < 9 \). Here the inequality is false.Therefore, for \( n = 2 \) and \( n = 3 \), the inequality is not true.

Step 3: Test Larger Positive Integers

To confirm the pattern, test larger values. Test \( n = 4 \):- \( 2^4 = 16 \) and \( 4^2 = 16 \), so \( 16 = 16 \).Now test \( n = 5 \):- \( 2^5 = 32 \) and \( 5^2 = 25 \), so \( 32 > 25 \).For \( n > 4 \), we notice \( 2^n \) exceeds \( n^2 \). Hence, only \( n = 2, 3, 4 \) are suspects.

Step 4: Verify for All Cases

Verify that for \( n > 4 \), \( 2^n > n^2 \). Observing the exponential growth of \( 2^n \), we see that as \( n \) increases, \( 2^n \) quickly surpasses \( n^2 \). Previously shown calculations confirm this trend. Thus, the inequality is not true only for \( n = 2, 3, 4 \).

Key Concepts

Positive Integers

Positive integers are whole numbers greater than zero. They form a subset of the set of natural numbers, which include zero as well.

These numbers can be represented as 1, 2, 3, 4, and so on, extending indefinitely. Positive integers are fundamental in understanding and solving inequalities. When approaching problems involving positive integers, consider the following key points:

• They do not include negative numbers or zero.
• Each positive integer is greater than all integers below it, showcasing a clear order.
• They are used to count objects, position, and natural occurrences.

In the context of inequalities like the one in this exercise, it's crucial to only consider values of \( n \) that are positive integers and test them individually.

Exponential Growth

Exponential growth describes a process where the quantity increases at a constant multiplicative rate over time. In mathematical terms, an exponential function can be represented as \( a^n \), where \( a \) is a constant base greater than one, and \( n \) is the positive integer exponent.

In the inequality \( 2^n > n^2 \), the term \( 2^n \) exhibits exponential growth. As \( n \) increases, \( 2^n \) grows exponentially because the base 2 is repeatedly multiplied by itself. Here are some characteristics of exponential growth:

• Starts slow but accelerates quickly, eventually overtaking polynomial functions like \( n^2 \).
• The rate of growth increases with time, making it faster than linear or quadratic growth.
• It's a crucial concept in areas like population dynamics, finance, and computer science.

The exercise highlights how exponential functions eventually surpass polynomial expressions, which is affirmed when testing larger values of \( n \). For \( n > 4 \), \( 2^n \) exceeds \( n^2 \), confirming the behavior of exponential growth.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (like addition and multiplication). These expressions can define functions and equations, providing a way to represent real-world situations mathematically.

Within this exercise, the expressions \( 2^n \) and \( n^2 \) are examples of algebraic expressions:

• \( 2^n \) is an exponential expression where 2 is the base, and \( n \) is the exponent.
• \( n^2 \) is a power expression, a specific type of polynomial where \( n \) is raised to the power of 2.

When solving inequalities that involve these expressions, it's important to compare their growth rates:

• The exponential expression \( 2^n \) grows much faster compared to the polynomial expression \( n^2 \) as \( n \) increases beyond small values.
• Identifying the point where the exponential function overtakes the polynomial is key in determining the range of solutions for \( n \).

Understanding these concepts helps to solve not just this inequality but a wide variety of problems in algebra involving exponential and polynomial expressions.