Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Determine the positive integer values of \(n\), where the arithmetic sequence \(a_{n}\) satisfies the inequality. $$a_{n} \geq 2, \text { where } a_{n}=5-\frac{1}{2} n$$

Step-by-Step Solution

Answer

The positive integer values of \( n \) are 1, 2, 3, 4, 5, and 6.

Step 1: Define the nth term

First, we express the nth term of the arithmetic sequence. We are given that for any term in the sequence, \( a_n = 5 - \frac{1}{2}n \).

Step 2: Set Up the Inequality

We want to find the integer values of \( n \) for which \( a_n \) satisfies the inequality \( a_n \geq 2 \). Substitute \( a_n \) into the inequality: \( 5 - \frac{1}{2}n \geq 2 \).

Step 3: Solve for n

Solve the inequality \( 5 - \frac{1}{2}n \geq 2 \). Subtract 5 from both sides to get \( -\frac{1}{2}n \geq -3 \). Multiply each side by -2 (remember to flip the inequality sign) to obtain \( n \leq 6 \).

Step 4: Determine the Positive Integer Values

Given the condition \( n \leq 6 \) and considering only positive integers, \( n \) can be 1, 2, 3, 4, 5, or 6.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values that are not equal. They are used to represent the relative size or order of values. In the context of this exercise, inequalities help determine the values for which the arithmetic sequence follows a specific condition, here being greater than or equal to 2.

When solving inequalities, similar strategies to solving equations can be used, such as adding or subtracting the same value from both sides. However, it's crucial to remember that when you multiply or divide both sides by a negative number, the inequality sign flips. For example,

• Adding a number: if you have \(x - 3 < 5\), you add 3 to get \(x < 8\).
• Multiplying by a negative: if \(-2x > 4\), dividing both sides by \(-2\) flips the inequality to \(x < -2\).

Understanding how to manipulate inequalities allows us to find all possible values that satisfy the inequality condition, thereby unlocking insights into the sequence’s behavior.

Term of a Sequence

A sequence is a list of numbers arranged in a specific order. These are often defined by a formula that tells you how to find each term in the sequence. For arithmetic sequences, such formulas involve a starting point and a pattern of adding or subtracting a consistent number.

In this exercise, the formula for the nth term of the sequence is given as \( a_n = 5 - \frac{1}{2}n \). This tells us how each term in the sequence relates to its position. The formula can be used to calculate the value of any term in the sequence. In arithmetic sequences:

• The constant we add or subtract (here \(\frac{1}{2}\)) is called the common difference.
• The term \(a_1\) is typically the first term of the sequence.

So, in this formula, reducing \(n\) by \(\frac{1}{2}\) suggests that each subsequent term is smaller by this fraction, implying a descending order. Such a formula can be applied to find which sequence terms meet specified conditions, like being over a certain value.

Positive Integers

Positive integers are the set of all positive whole numbers that begin from 1 and increase indefinitely (1, 2, 3, and so on). They form the part of the integers under consideration when constraints like positivity are applied to mathematical problems.

In the given exercise, we're tasked with finding positive integer values of \(n\) that satisfy certain conditions. Positive integers are critical because they ensure the sequence is explored using feasible and meaningful positions.

• They do not include zero or any negative numbers.
• When dealing with terms of a sequence, these integers denote the position of each element uniquely.

Since the solution to the inequality resulted in \(n \leq 6\), checking only positive integer values gives us real-world applicable results that make mathematical sense within the problem's context. Thus, 1, 2, 3, 4, 5, and 6 are the potential values for \(n\) that satisfy both the inequality and the condition of positivity.