Problem 6 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The sequence's values are \(|a_0| = 1\), \(|a_1| = \frac{1}{\sqrt{2}}\), \(|a_2| = \frac{1}{\sqrt{3}}\), \(|a_3| = \frac{1}{2}\), \(|a_4| = \frac{1}{\sqrt{5}}\), and \(|a_5| = \frac{1}{\sqrt{6}}\).

Step 1: Understand the Sequence Formula

The sequence is described by the formula \(a_{n} = \frac{1}{\sqrt{n+1}}\). This means for each \(n\), we substitute \(n\) into the formula to find the terms of the sequence.

Step 2: Calculate \\(a_0\\)

Substitute \(n = 0\) into the formula: \(a_0 = \frac{1}{\sqrt{0+1}} = \frac{1}{1} = 1\). Thus, \(|a_0| = 1\).

Step 3: Calculate \\(a_1\\)

Substitute \(n = 1\) into the formula: \(a_1 = \frac{1}{\sqrt{1+1}} = \frac{1}{\sqrt{2}}\). Hence, \(|a_1| = \frac{1}{\sqrt{2}}\).

Step 4: Calculate \\(a_2\\)

Substitute \(n = 2\) into the formula: \(a_2 = \frac{1}{\sqrt{2+1}} = \frac{1}{\sqrt{3}}\). Thus, \(|a_2| = \frac{1}{\sqrt{3}}\).

Step 5: Calculate \\(a_3\\)

Substitute \(n = 3\) into the formula: \(a_3 = \frac{1}{\sqrt{3+1}} = \frac{1}{\sqrt{4}} = \frac{1}{2}\). Therefore, \(|a_3| = \frac{1}{2}\).

Step 6: Calculate \\(a_4\\)

Substitute \(n = 4\) into the formula: \(a_4 = \frac{1}{\sqrt{4+1}} = \frac{1}{\sqrt{5}}\). Hence, \(|a_4| = \frac{1}{\sqrt{5}}\).

Step 7: Calculate \\(a_5\\)

Substitute \(n = 5\) into the formula: \(a_5 = \frac{1}{\sqrt{5+1}} = \frac{1}{\sqrt{6}}\). Thus, \(|a_5| = \frac{1}{\sqrt{6}}\).

Key Concepts

Sequence Evaluation

A sequence is a set of numbers arranged in a specific order following a distinct rule. Evaluating a sequence means determining its terms based on the given rule. In this exercise, the sequence is represented by \(a_n = \frac{1}{\sqrt{n+1}}\). To find the various terms of this sequence, we substitute different values of \(n\) into the sequence formula. This approach helps us determine the value of each term, starting from \(n = 0\) up to \(n = 5\).

This specific sequence describes an infinite series of terms where each term is influenced by its position \(n\). Here are the first few evaluations for better understanding:

• For \(n = 0\), substitute into \(a_n\) to get \(a_0 = 1\).
• Continue substituting subsequent values, for \(n = 1, 2, 3, 4, 5\) to obtain the sequence terms as specified in the exercise.

This methodical process allows us to discover and understand each term and their absolute values, which plays a key role in more complex mathematical reasoning.

Absolute Value

The absolute value of a number is essentially its distance from zero on a number line, without considering its direction. Defined as \(|a|\), it is always a non-negative value. When dealing with sequences, it becomes pertinent to consider the absolute value of each term, especially when the terms might be negative, although in this specific case they are not.

For this particular sequence, each individual term \(a_n\) is derived based on the formula \(\frac{1}{\sqrt{n+1}}\). When calculating the absolute value, we simply take the non-negative value of each evaluated term. In the exercise example:

• \(|a_0| = 1\)
• \(|a_1| = \frac{1}{\sqrt{2}}\)
• \(|a_2| = \frac{1}{\sqrt{3}}\)

And so on...

This ensures that each result remains non-negative, providing a consistent sequence of values.

Square Root Calculation

The square root is a mathematical function that finds a number which, when multiplied by itself, gives the original number. Represented by the radical symbol \(\sqrt{}\), it can affect calculations significantly by changing the behavior of the values it transforms.

In this sequence exercise, each term depends on the square root of \(n + 1\) found in the denominator \(\frac{1}{\sqrt{n+1}}\). Calculating the square root is a crucial step in determining the sequence. Let's explore how it is applied:

• For \(n = 2\), calculate \(\sqrt{3}\) to find \(a_2 = \frac{1}{\sqrt{3}}\).
• For \(n = 4\), find \(\sqrt{5}\) to determine \(a_4 = \frac{1}{\sqrt{5}}\).

This process is repeated for each term. Understanding how to calculate square roots and apply them helps simplify the expression, ultimately solving parts of many calculations such as those found in this sequence problem.