Problem 3 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\(a_0 = \frac{1}{2}, a_1 = \frac{1}{3}, a_2 = \frac{1}{4}, a_3 = \frac{1}{5}, a_4 = \frac{1}{6}, a_5 = \frac{1}{7}\).

Step 1: Understand the Sequence Formula

The sequence is given by the formula: \(a_{n}=\frac{1}{n+2}\), where \(n\) is a non-negative integer (\(n = 0, 1, 2, \ldots\)). This formula will be used to calculate the value of \(a_n\) for given values of \(n\).

Step 2: Calculate \(a_{0}\)

Substitute \(n = 0\) into the sequence formula: \(a_{0} = \frac{1}{0+2} = \frac{1}{2}\).

Step 3: Calculate \(a_{1}\)

Substitute \(n = 1\) into the sequence formula: \(a_{1} = \frac{1}{1+2} = \frac{1}{3}\).

Step 4: Calculate \(a_{2}\)

Substitute \(n = 2\) into the sequence formula: \(a_{2} = \frac{1}{2+2} = \frac{1}{4}\).

Step 5: Calculate \(a_{3}\)

Substitute \(n = 3\) into the sequence formula: \(a_{3} = \frac{1}{3+2} = \frac{1}{5}\).

Step 6: Calculate \(a_{4}\)

Substitute \(n = 4\) into the sequence formula: \(a_{4} = \frac{1}{4+2} = \frac{1}{6}\).

Step 7: Calculate \(a_{5}\)

Substitute \(n = 5\) into the sequence formula: \(a_{5} = \frac{1}{5+2} = \frac{1}{7}\).

Key Concepts

Sequence Formula

A sequence is a list of numbers following a specific pattern or rule. In mathematics, the sequence formula is the equation that defines the sequence by providing a rule for determining each term. For example, in the sequence given by the formula \( a_n = \frac{1}{n+2} \), \( n \) represents a non-negative integer (0, 1, 2, etc.).
The formula serves as a blueprint, showing how to calculate each term in the sequence. It is a powerful tool that allows you to find any term in the sequence without needing to know the previous one. This is especially useful for sequences where direct computation is more efficient than iterative counting through previous values. To apply a sequence formula:

• Identify \( n \), the position of the term in the sequence you wish to find.
• Substitute \( n \) into the formula.
• Perform the necessary arithmetic operations to get the term's value.

Utilizing the correct formula is crucial as it directly determines the properties and behavior of the sequence.

Calculation of Terms

Calculating terms in a sequence involves substituting different values of \( n \) (often starting from 0 or 1) into the sequence formula and evaluating the expression. This process allows you to find the exact value of each term, which in our specific sequence \( a_n = \frac{1}{n+2} \), is quite straightforward.
Let's illustrate this with a few examples:

• For \( n = 0 \), substitute into the formula: \( a_0 = \frac{1}{0+2} = \frac{1}{2} \). This means the first term of the sequence is \( \frac{1}{2} \).
• For \( n = 1 \), \( a_1 = \frac{1}{1+2} = \frac{1}{3} \), indicating the second term is \( \frac{1}{3} \).
• Continue this method for subsequent values: \( a_2 = \frac{1}{4} \), \( a_3 = \frac{1}{5} \), and so on.

The goal is to understand that each term’s calculation involves simple arithmetic once the formula is known. This systematic approach ensures that any term, regardless of its position in the sequence, can be accurately determined.

Finite Sequences

A finite sequence is one that has a definite number of terms. This contrasts with infinite sequences, which continue indefinitely without an endpoint. In the case of finite sequences, such as the one provided where \( n = 0, 1, 2, \ldots, 5 \), the sequence only contains the terms up to when \( n = 5 \).
Understanding finite sequences is particularly important as they often appear in practical applications where you only need a limited number of values in a sequence.
When dealing with finite sequences:

• The sequence starts and ends at specific positions, known as the initial and final terms.
• Finite sequences can be easily represented and analyzed since they consist of countable terms.
• Upon reaching the final term, the sequence is considered complete.

Finite sequences are simpler to handle since you do not have to concern yourself with terms beyond the specified endpoint, making them easier to use in analytical discussions or specific mathematical applications.