Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Determine the positive integer values of \(n\), where the sequence \(a_{n}\) satisfies the inequality. $$a_{n}<11, \text { where } a_{n}=n^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Sequence

We are given the sequence \(a_n = n^2\). The term \(a_n\) simply represents the square of the positive integer \(n\).

Step 2: Set Up the Inequality

Our task is to find values of \(n\) such that \(a_n < 11\) holds true. Thus, we need to solve the inequality: \(n^2 < 11\).

Step 3: Solve the Inequality

We need to find positive integer values of \(n\) satisfying \(n^2 < 11\). Calculate the square root of both sides to get \(n < \sqrt{11}\). Since \(\sqrt{11} \approx 3.316\), the possible positive integer values for \(n\) are below 3.316.

Step 4: Identify Integer Values of \(n\)

The integers less than 3.316 are 1, 2, and 3. We confirm each:- For \(n = 1\), \(1^2 = 1 < 11\)- For \(n = 2\), \(2^2 = 4 < 11\)- For \(n = 3\), \(3^2 = 9 < 11\)All these satisfy the inequality.

Key Concepts

Quadratic Sequences

A quadratic sequence is an ordered number list in which the difference between consecutive terms changes at a constant rate. This characteristic makes quadratic sequences distinct from arithmetic or geometric sequences. The general form of a quadratic sequence is given as \(a_n = an^2 + bn + c\), where \(a, b\), and \(c\) are constants, and \(n\) represents the term number.
In this exercise, the sequence is characterized by \(a_n = n^2\), which is a simplified quadratic sequence where \(a = 1, b = 0,\) and \(c = 0\). This means each term in the sequence is simply the square of its position number:

• For \(n=1\), the sequence term is \(1^2 = 1\)
• For \(n=2\), the sequence term is \(2^2 = 4\)
• And for \(n=3\), the sequence term is \(3^2 = 9\)

Recognizing that these sequences can never become negative helps form a foundation for solving inequalities associated with them, as in this case when we analyze \(n^2 < 11\).

Integer Solutions

Integer solutions refer to solutions that are whole numbers, both positive or negative, and zero. However, in this context, only positive integer solutions are considered for the variable \(n\). When tackling problems like the one provided, identifying integer solutions is crucial because real-life applications often demand whole numbers.
To find integer solutions for \(n\) in the inequality \(n^2 < 11\), we must first consider the square root of 11. The value of \( \sqrt{11} \) is approximately 3.316. Thus, \(n\) should be less than 3.316 to satisfy the inequality with integers included. This means that the set of possible \(n\) values include:

• \(n = 1\), as \(1^2 = 1 < 11\)
• \(n = 2\), since \(2^2 = 4 < 11\)
• \(n = 3\), because \(3^2 = 9 < 11\)

All these positive integers meet the criteria of being an integer solution while also respecting the condition \(n^2 < 11\).

Sequence Inequalities

Solving sequence inequalities involves finding the range of numbers that satisfy a given inequality condition. This involves inequalities where the sequence terms must maintain a constraint, such as being less than or greater than a specific value.
In the presented inequality \(a_n < 11\) for the sequence \(a_n = n^2\), solving it requires finding terms in the sequence that are less than 11. The process includes:

• Setting up the inequality \(n^2 < 11\)
• Taking the square root of both sides to find \(n < \sqrt{11}\)
• Calculating \(\sqrt{11} \approx 3.316\) and determining integer values less than this, which are \(n = 1, 2, 3\)

This approach ensures that all possible terms in the sequence comply with the restriction given by the inequality, allowing only those terms that truly fit within the condition to be considered valid solutions. Pulling specific numbers from non-integer results into integers requires careful approximation and understanding of where those values fall on the number line.