Problem 11 - Calculus for Biology and Medicine

Question

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{n^{2}}{n+1} $$

Step-by-Step Solution

Answer

The values are 0, \(\frac{1}{2}\), \(\frac{4}{3}\), \(\frac{9}{4}\), \(\frac{16}{5}\), and \(\frac{25}{6}\).

Step 1: Understand the Sequence Formula

Given the sequence where the terms are defined by the formula \(a_n = \frac{n^2}{n+1}\). We want the absolute values of these terms for \(n = 0, 1, 2, \ldots, 5\). The absolute value of \(a_n\) is simply \(|a_n|\) since \(a_n\) is always positive in this range.

Step 2: Calculate the First Term

Substitute \(n = 0\) into the sequence formula: \[ a_0 = \frac{0^2}{0+1} = \frac{0}{1} = 0 \]Thus, \(|a_0| = 0\).

Step 3: Calculate the Second Term

Substitute \(n = 1\) into the formula: \[ a_1 = \frac{1^2}{1+1} = \frac{1}{2} \]Thus, \(|a_1| = \frac{1}{2}\).

Step 4: Calculate the Third Term

For \(n = 2\),\[ a_2 = \frac{2^2}{2+1} = \frac{4}{3} \]Thus, \(|a_2| = \frac{4}{3}\).

Step 5: Calculate the Fourth Term

Substitute \(n = 3\):\[ a_3 = \frac{3^2}{3+1} = \frac{9}{4} \]Thus, \(|a_3| = \frac{9}{4}\).

Step 6: Calculate the Fifth Term

For \(n = 4\),\[ a_4 = \frac{4^2}{4+1} = \frac{16}{5} \]Thus, \(|a_4| = \frac{16}{5}\).

Step 7: Calculate the Sixth Term

Finally, substitute \(n = 5\):\[ a_5 = \frac{5^2}{5+1} = \frac{25}{6} \]Thus, \(|a_5| = \frac{25}{6}\).

Key Concepts

Absolute Value in Sequences

Understanding the concept of absolute value is crucial in various areas of mathematics, including sequences in calculus. The absolute value of a number, denoted by the vertical bars \(|x|\), represents the distance of that number from zero on a number line. Therefore, the absolute value is always a non-negative number.

For any real number \(x\), the absolute value is defined as:
- \( |x| = x \) if \( x \geq 0 \) - \( |x| = -x \) if \( x < 0 \)

In the context of sequences, when evaluating \(|a_n|\) as seen in the exercise, this definition ensures that each term of the sequence is considered in its non-negative form. In this particular example, since all evaluated terms \(a_n\) for \(n=0, 1, 2, \ldots, 5\) are positive, \(|a_n| = a_n\). This simplifies our calculations as we don't need to adjust the sign of any terms.

Sequence Formula in Calculus

Sequences are ordered lists of numbers defined by a specific formula or rule. In calculus, understanding sequence formulas helps in analyzing patterns and behaviors of sequences over ranges or indefinitely.

The given sequence formula \(a_n = \frac{n^2}{n+1}\) illustrates how each term can be generated based on its position \(n\) in the sequence. Here, \(n\) indicates the term's position starting from zero. To determine each term, simply substitute the desired value of \(n\) into the formula, perform the arithmetic operations, and obtain the result.

Analyzing this sequence, note how the numerator grows faster (since \(n^2\) grows quicker than \(n\)), influencing the overall behavior as you move through the sequence. Studying this allows us to predict that \(a_n\) will increase as \(n\) increases, reflecting the behavior you see in each calculated value for \(n=0, 1, 2, \ldots, 5\).

Mathematical Problem Solving Techniques

Effective problem solving in mathematics involves a structured approach:

• Understand the Problem: Carefully read and interpret the problem to understand what is being asked. For sequences, this often means grasping the formula itself and what values need evaluation.
• Devise a Plan: Outline the steps needed. Here, the plan involves substituting consecutive values of \(n\) into the sequence formula and computing results, followed by taking their absolute values.
• Carry Out the Plan: Execute the steps methodically. Substitution into the sequence formula yields terms like \(a_0 = 0\) and \(a_1 = \frac{1}{2}\).
• Review/Reflect: Check results for correctness. This entails comparing computed \(|a_n|\) values to expected results or interpreting their significance, ensuring calculations align with mathematical expectations.

By following these steps, not only do you ensure accuracy, but you also develop deeper insights into the problem-solving process, a crucial skill in calculus and beyond.