Problem 85 - A Graphical Approach to Precalculus with Limits

Question

Determine the positive integer values of \(n\), where the sequence \(a_{n}\) satisfies the inequality. $$a_{n}<10, \text { where } a_{n}=n+1$$

Step-by-Step Solution

Answer

The positive integer values of \(n\) are 1, 2, 3, 4, 5, 6, 7, and 8.

Step 1: Understand the Sequence Definition

The sequence given is defined as \(a_n = n + 1\). This means for any integer \(n\), the term in the sequence is simply \(n\) plus 1.

Step 2: Set Up the Inequality

We are tasked with finding the integer values of \(n\) such that the sequence \(a_n\) is less than 10. Thus, the inequality we need to satisfy is \(n + 1 < 10\).

Step 3: Solve the Inequality

Do some algebraic manipulation by subtracting 1 from both sides of the inequality to isolate \(n\) on one side:\[ n + 1 < 10 \]\[ n < 10 - 1 \]\[ n < 9 \]

Step 4: Determine Positive Integer Solutions

The solution to the inequality \(n < 9\) tells us that \(n\) can be any positive integer less than 9. Since \(n\) is a positive integer, the possible solutions are 1, 2, 3, 4, 5, 6, 7, and 8.

Key Concepts

Sequence Definition

In mathematics, a sequence is a set of numbers arranged in a specific order. For the problem at hand, we have a sequence defined by the expression \(a_n = n + 1\). This means that each term in the sequence is generated by taking an integer \(n\) and adding 1 to it.
To understand the sequence better, consider an example:

• If \(n = 1\), then \(a_1 = 1 + 1 = 2\).
• If \(n = 2\), then \(a_2 = 2 + 1 = 3\).
• If \(n = 3\), then \(a_3 = 3 + 1 = 4\).

This is how each term develops based on the formula \(a_n = n + 1\). Understanding the structure of this sequence is crucial because it defines how we interpret and solve the associated inequality.

Integer Solution

In the context of this exercise, solving the inequality \(n + 1 < 10\) for integer solutions means finding all whole numbers (no fractions or decimals) that satisfy this condition.
by definition, integers are values like -2, -1, 0, 1, 2, and so on. However, the problem specifically asks for positive integers, so we only consider integers greater than zero.
Using algebraic manipulation, we simplified the inequality to \(n < 9\). This suggests that \(n\) can take any integer value less than 9. Since \(n\) must be positive, we list out these possibilities: 1, 2, 3, 4, 5, 6, 7, and 8. It's important to remember these values as they are the solution set.Listing this range of solutions clearly highlights the understanding of integer values within the constraints of the inequality, ensuring our answer is both accurate and thorough.

Algebraic Manipulation

Algebraic manipulation is a process used to simplify or solve equations and inequalities. It involves rearranging expressions and performing operations to isolate variables. In our problem, we start with the inequality: \(n + 1 < 10\).To isolate \(n\), we perform a simple algebraic operation: subtracting 1 from both sides of the inequality. This gives us:

• Subtract 1 from left: \(n + 1 - 1 = n\)
• Subtract 1 from right: \(10 - 1 = 9\)

After manipulation, the inequality becomes \(n < 9\). By isolating \(n\), we can more clearly see the range of values that satisfy the inequality. This manipulation is crucial because it allows us to focus purely on the variable of interest (\(n\) in this case) when determining which numbers meet the criteria of the problem.

Positive Integer Values

When determining solutions to inequalities like \(n < 9\) in our sequence context, only positive integer values matter here. Positive integers are numbers like 1, 2, 3, and so on, extending upwards in value. Given the solution of our inequality \(n < 9\), we have to consider the constraint that \(n\) must be positive. Therefore, we exclude 0 and any negative numbers, focusing instead on 1 through 8 as valid solutions.
The positive integers less than 9 are:

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8

Each of these numbers, when substituted for \(n\), makes the sequence term \(a_n = n + 1\) strictly less than 10, satisfying the original condition.