Problem 22 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The values are \( n = 1, 2, 3 \).

Step 1: Understanding the Problem

We need to find positive integers \( n \) such that \( n! \) is not greater than \( 2n \). This means \( n! \leq 2n \).

Step 2: Testing Small Values of n

Start with small positive integers and calculate \( n! \) and compare it to \( 2n \).- For \( n = 1 \): \( 1! = 1 \) and \( 2 \times 1 = 2 \), so \( 1! \leq 2 \times 1 \).- For \( n = 2 \): \( 2! = 2 \) and \( 2 \times 2 = 4 \), so \( 2! \leq 2 \times 2 \).- For \( n = 3 \): \( 3! = 6 \) and \( 2 \times 3 = 6 \), so \( 3! = 2 \times 3 \).

Step 3: Checking for Larger Values of n

Now, check values greater than 3.- For \( n = 4 \): \( 4! = 24 \) and \( 2 \times 4 = 8 \), so \( 4! > 2 \times 4 \).- For \( n > 4 \): Factorials grow rapidly, so \( n! \) will be much larger than \( 2n \).

Step 4: Concluding the Solution

The inequality \( n! \leq 2n \) holds only for \( n = 1, 2, 3 \) as larger numbers result in \( n! \) being much greater than \( 2n \). Therefore, these are the values for which the statement is not true.

Key Concepts

Positive Integers

Positive integers, often referred to as the natural numbers, play a crucial role in various areas of mathematics, including solving inequalities. These are numbers that begin at 1 and increase without end. When dealing with factorial inequalities, such as in our problem, we're specifically interested in the properties of these numbers and how they behave under factorial and linear expressions.

In the context of our problem, we are asked to focus on positive integers for which the factorial operation, denoted by "!", does not result in a value greater than double the integer. The fascinating aspect of positive integers is their predictable, sequential growth, which plays into understanding how quickly factorials can grow compared to linear expressions. This fundamental property can help us determine solutions effectively.

• Positive integers: 1, 2, 3, 4, ...
• No fractions or decimals
• Used commonly in counting and order

Comparing Factorial and Linear Expressions

Comparing factorial and linear expressions is at the heart of understanding why only certain small integers satisfy the inequality \( n! \leq 2n \). A factorial, represented by \( n! \), is a product of all positive integers up to \( n \). It's a rapidly growing function.

On the other hand, a linear expression like \( 2n \) increases steadily in comparison. This slow and constant growth is what makes the comparison so revealing; it highlights how quickly factorial expressions outpace linear ones as \( n \) increases.

• \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \)
• \( 2n \) simply equals two times the integer \( n \)
• Factorials grow exponentially, not linearly
• After certain values, \( n! \) becomes substantially larger than \( 2n \) due to compounded multiplication

By testing small values of \( n \), as outlined in our problem, the exponential growth of factorials becomes evident, confirming that only the smallest positive integers comply with the inequality.

Mathematical Problem Solving

Solving mathematical problems, such as factorial inequalities, requires a methodical approach. This involves understanding the problem, trying out possibilities, and identifying patterns. In this particular case, the problem asked us to determine under what conditions the factorial of a number is not greater than twice that number.

The solution process begins with testing small, manageable values of \( n \). Observing results for \( n = 1, 2, \) and \( 3 \) gives us direct insight as to where the inequality holds. The testing phase is crucial in mathematical problem-solving as it either proves or disproves initial hypotheses. Once we reached \( n = 4 \), the factorial expression \( 4! \) surpassed twice the number \( 2 \times 4 \) by a wide margin. This confirms the end of the solutions due to factorial growth.

• Recognize and understand the core problem
• Test various cases to uncover patterns
• Identify critical points where the behavior changes
• Leverage mathematical properties and structures

This structured approach often reveals underlying principles not only applicable in this problem but also in a wide array of mathematical endeavours.