Problem 19 - A Graphical Approach to College Algebra

Question

Find all positive integers \(n\) for which the given statement is not true. $$3^{n}>6 n$$

Step-by-Step Solution

Answer

The inequality does not hold for \( n = 1 \) and \( n = 2 \).

Step 1: Understanding the Problem

We need to find positive integer values of \( n \) for which the inequality \( 3^n > 6n \) does not hold.

Step 2: Set up the Inequality

We start with the expression \( 3^n \) and want to find when it is not greater than \( 6n \). Thus, we are solving for \( 3^n \leq 6n \).

Step 3: Test Small Values of n

To find possible integer \( n \), let's test the values of \( n = 1, 2, 3, 4, \) and so on to see when \( 3^n \leq 6n \).

Step 4: Calculate for n = 1

For \( n = 1 \), \( 3^1 = 3 \) and \( 6 imes 1 = 6 \). Since \( 3 \leq 6 \), this condition holds.

Step 5: Calculate for n = 2

For \( n = 2 \), \( 3^2 = 9 \) and \( 6 imes 2 = 12 \). Since \( 9 \leq 12 \), this condition holds.

Step 6: Calculate for n = 3

For \( n = 3 \), \( 3^3 = 27 \) and \( 6 imes 3 = 18 \). Since \( 27 > 18 \), this condition does not hold.

Step 7: General Observation

After reviewing the calculations, we observe that for \( n \geq 3 \), \( 3^n \) grows much faster than \( 6n \), which means that \( 3^n > 6n \) for these values. Therefore, it is likely that only \( n = 1 \) and \( n = 2 \) will satisfy \( 3^n \leq 6n \).

Step 8: Verification Step

To verify, let's analyze the inequality analytically. For \( 3^n \leq 6n \), the ratio \( \frac{3^n}{n} \) should be less than or equal to 6. As \( n \) increases, \( \frac{3^n}{n} \) increases exponentially, confirming our earlier observation.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a consistent percentage or ratio over equal time intervals. This type of growth can quickly outpace linear growth, which increases by a constant amount over each interval. In the context of our problem, we deal with exponential growth through the term \(3^n\). As \(n\) increases, \(3^n\) grows significantly faster compared to linear functions like \(6n\). This is because each increase in \(n\) results in the value of \(3^n\) multiplying by 3, leading to rapid escalation.

Here are a few points to better understand exponential growth:

• The base number, in this case, 3, determines the growth rate per unit increment of \(n\).
• Exponential functions have distinctive properties, such as doubling over regular intervals, which does not occur in linear growth.
• Exponential growth is non-linear and becomes dramatic as \(n\) increases, which is evident when compared to any linear equation such as \(6n\).

Due to these properties, when analyzing exponential growth in problems such as this inequality, it’s crucial to understand how quickly such functions grow compared to others like linear equations.

Integer Solutions

Integer solutions refer to solving equations or inequalities where the solutions must be whole numbers, without fractions or decimals. In algebraic problems, finding integer solutions can be critical when the problem constraints necessitate counting whole items or quantities.

In our inequality \(3^n \leq 6n\), we are tasked with identifying such integer values for \(n\). By testing consecutive values, starting from small numbers, we can establish a pattern or find specific numbers that fit within the problem's conditions.

• For \(n=1\), \(3^1=3\) and \(6 \times 1=6\), meaning the statement holds true for this integer.
• Similarly, testing other integers like \(n=2\) can help verify additional solutions.
• The approach typically involves verifying each integer step-by-step until the condition no longer holds.

This systematic process allows us to determine precisely for which integers the inequality or equation is true or false, which is fundamental in discrete math problems where only whole number solutions are realistic or relevant.

Algebraic Inequality

An algebraic inequality expresses a relationship where two algebraic expressions are not equal, showing that one is either less than, greater than, or not equal to the other. It is a type of inequality involving variables and constants. In our scenario, we explore the inequality \(3^n > 6n\) and its reverse, \(3^n \leq 6n\), to find where this does not hold.

Understanding and solving inequalities rely on several concepts:

• You can manipulate inequalities similarly to equations through addition, subtraction, multiplication, and division, but be cautious when multiplying or dividing by negative numbers as this reverses the inequality sign.
• Setting up initial conditions, like testing small values, helps establish a pattern or boundary for when the inequality changes.
• Verification through numerical examples or analytical methods, such as ratios, further confirms which values satisfy or do not satisfy the inequality.

While solving the inequality \(3^n \leq 6n\), we initially focus on intuition by calculating for small values of \(n\) and observing when the inequality flips, and later verify the observations analytically.