Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality. .$$a_{n} \geq \frac{2}{27}, \text { where } a_{n}=\frac{6}{3^{n}}$$

Step-by-Step Solution

Answer

Positive integer values of \( n \) are 1, 2, 3, and 4.

Step 1: Setting Up the Inequality

We start by substituting the expression for the geometric sequence into the inequality. The sequence is given by \( a_n = \frac{6}{3^n} \). Therefore, the inequality becomes: \[ \frac{6}{3^n} \geq \frac{2}{27}. \] This is our inequality to solve for \( n \).

Step 2: Isolating the Exponential Term

To isolate the exponential term, we multiply both sides of the inequality by \( 3^n \) to eliminate the fraction: \[ 6 \geq \frac{2}{27} \times 3^n. \] This simplifies to: \[ 6 \geq \frac{2 \cdot 3^n}{27}. \]

Step 3: Simplifying the Right Side

We simplify the right side of the inequality by reducing the fraction: \[ \frac{2 \cdot 3^n}{27} = \frac{2}{27} \cdot 3^n. \] Thus, the inequality becomes: \[ 6 \geq \frac{2}{27} \times 3^n. \] Our goal is to solve for \( n \).

Step 4: Multiplying to Eliminate the Denominator

Multiply both sides by 27 to clear the fraction: \[ 162 \geq 2 \cdot 3^n. \] This simplifies further to: \[ 162 = 2 \cdot 3^n. \]

Step 5: Solving for the Exponent

Divide both sides by 2 to isolate the exponential term: \[ 81 \geq 3^n. \] Now we need to solve for \( n \).

Step 6: Evaluating Powers of 3

Recognize powers of 3: \( 3^0 = 1, 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243 \). We identify that \( 3^4 = 81 \). Therefore, \( 3^n \leq 3^4 \).

Step 7: Determining Values of n

Since \( 3^n \leq 81 \), the values of \( n \) are such that \( n \leq 4 \). Since we seek positive integers, \( n = 1, 2, 3, 4 \).

Key Concepts

Inequalities

Inequalities are a mathematical way to show the relationship between two different values. They help us compare whether one number is greater than, less than, or equal to another number. These can be expressed using the symbols:

• \( > \) means greater than.
• \( < \) means less than.
• \( \geq \) means greater than or equal to.
• \( \leq \) means less than or equal to.

In the context of geometric sequences, inequalities are used to determine which terms in the sequence satisfy a certain condition.
For example, determining which term of a sequence is at least a specific value.

When solving an inequality, you often perform similar steps as solving an equation, such as adding, subtracting, multiplying, or dividing both sides by the same number.
However, remember if you multiply or divide an inequality by a negative number, the inequality sign flips direction.

Exponential Equations

Exponential equations are types of equations where the variable is in the exponent. These involve expressions like \( b^n \), where \( b \) is the base and \( n \) is the exponent or power.
In these equations, you are usually solving for the exponent.

• A common approach to solving exponential equations is making the bases on both sides of the equation the same.
• Once the bases are equal, you can set the exponents equal to each other.

In the original exercise, the geometric sequence forms part of an exponential equation.
We isolated the exponential term to solve for the variable \( n \).
This highlights a common approach when dealing with such equations, especially in geometric sequences where terms decrease or increase exponentially based on the exponent.

Solving Inequalities

Solving inequalities involves finding the set of all possible values for a variable that make the inequality true. There are steps similar to solving equations but with careful handling, especially when multiplying or dividing by negative numbers.
In the practice exercise:

• We started by expressing the inequality with the terms of the geometric sequence.
• The goal was to find meet the condition \( a_n \geq \frac{2}{27} \).
• By multiplying to remove fractions and dividing to isolate the exponential term, we simplify the inequality progressively.

Once the inequality is simplified to a comparison of powers, as exemplified by \( 81 \geq 3^n \), recognizing familiar powers of numbers helps.

Finally, by comparing, we discovered the range of possible positive integer solutions for \( n \), eq., \( 1 \leq n \leq 4 \).
This teaches not only calculation skills but also how to approach and simplify inequalities, getting to manageable terms where direct comparison leads to solutions.