Problem 1 - Calculus for Biology and Medicine

Question

In Problems \(1-16\), determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=n $$

Step-by-Step Solution

Answer

The sequence values are 0, 1, 2, 3, 4, 5.

Step 1: Understand the Sequence Definition

The sequence is defined by the expression \( a_n = n \). This means that each term in the sequence is equal to its index. For example, \( a_0 \) is the value when \( n = 0 \), \( a_1 \) is the value when \( n = 1 \), and so on.

Step 2: Compute Values for Each Term

We compute the values one by one from \( n = 0 \) to \( n = 5 \). For each \( n \), use the formula \( a_n = n \). - For \( n = 0 \), \( a_0 = 0 \).- For \( n = 1 \), \( a_1 = 1 \).- For \( n = 2 \), \( a_2 = 2 \).- For \( n = 3 \), \( a_3 = 3 \).- For \( n = 4 \), \( a_4 = 4 \).- For \( n = 5 \), \( a_5 = 5 \).

Step 3: List the Sequence Values

Now, organize the computed values into a list to represent the sequence: \( a_0 = 0, a_1 = 1, a_2 = 2, a_3 = 3, a_4 = 4, a_5 = 5 \). So, the sequence values for \( n = 0, 1, 2, 3, 4, 5 \) are \( 0, 1, 2, 3, 4, 5 \).

Key Concepts

Sequence Definition

A sequence is an ordered list of numbers, organized in a specific pattern. When we talk about sequences in mathematics, each number in the pattern is called a term.
For a clearer understanding, think of a sequence as a set of numbers assigned based on a rule or formula, which we call a sequence expression. This concept is crucial in analyzing patterns and doing advanced math studies.

• A sequence can have a finite number of terms or go on indefinitely (infinite).
• The rule defining the sequence tells us how to find each term based on its position or index.

In our example, the sequence is given by the expression \( a_n = n \), meaning the value of the sequence at any position \( n \) is just \( n \). This very simple sequence directly uses index numbers, helping to illustrate how sequences are not always complex.

Sequence Computation

Understanding how to compute a sequence involves applying the given rule to discover each term's value based on its index.
To compute the values of our sequence \( \{a_n\} \) where \( a_n = n \), you perform the following steps:

• Identify the range of indices \( n \) to compute. Here, it is from 0 to 5.
• For each value of \( n \), apply the formula \( a_n = n \) to find the term.

Let's see this in action:
\[\begin{align*} n = 0, & \quad a_0 = 0 \ n = 1, & \quad a_1 = 1 \ n = 2, & \quad a_2 = 2 \ n = 3, & \quad a_3 = 3 \ n = 4, & \quad a_4 = 4 \ n = 5, & \quad a_5 = 5 \ \end{align*}\] By systematically applying the sequence definition, we uncover each value quickly and clearly.

Index Notation

Index notation in sequences helps us identify and describe each term's position efficiently. The position or order of a term is called its index, often shown as a variable like \( n \).

• Each term in the sequence is marked by the index notation \( a_n \), where \( n \) specifies the term's position.
• In the expression \( a_n = n \), the subscript \( n \) indicates where in the sequence this term belongs.

The use of index notation is particularly important in conveying the sequence's organization without writing out all terms explicitly. It's a powerful way to define sequences briefly and to communicate their structure effectively.