Problem 28 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ 1,3,5,7,9, \ldots $$

Step-by-Step Solution

Answer

The expression for \(a_n\) is \(a_n = 1 + 2n\).

Step 1: Observe the sequence

First, observe the given sequence: 1, 3, 5, 7, 9, .... It is an arithmetic sequence where each number increases by a common difference.

Step 2: Identify the first term and common difference

In this sequence, the first term, denoted as \(a_0\), is 1 and the common difference \(d\) is 2. This is because each term increases by 2 from the previous one.

Step 3: Use the formula for an arithmetic sequence

The general formula for the \(n\)-th term of an arithmetic sequence is \(a_{n} = a_{0} + n imes d\), where \(a_0\) is the first term and \(d\) is the difference between terms.

Step 4: Substitute known values into the formula

Substitute \(a_0 = 1\) and \(d = 2\) into the formula: \(a_n = 1 + n imes 2\).

Step 5: Simplify the expression

Simplify the expression to find the formula for \(a_n\): \[a_n = 1 + 2n\].

Key Concepts

Understanding Common Difference

In arithmetic sequences, each term after the first is obtained by adding a constant number to the previous term. This constant is known as the "common difference." The common difference, often denoted by \(d\), is a fundamental part of arithmetic sequences.
The sequence given in the exercise is 1, 3, 5, 7, 9, ... . If we inspect the sequence, each number increases by the same fixed amount. Let's see how this works:

• The difference between 3 and 1 is 2.
• The difference between 5 and 3 is again 2.
• This pattern continues for the subsequent terms as well.

Therefore, the common difference \(d\) for this sequence is 2. This means that each term (except the first) is 2 more than the term before it. This consistent difference is what makes the sequence arithmetic.

Analyzing the Sequence

Sequence analysis involves understanding the structure and pattern of a sequence. For arithmetic sequences, like the one given in the exercise, each term is an addition of the common difference to the preceding term.
Analyzing this sequence, 1, 3, 5, 7, 9, ..., we notice:

• The sequence begins at 1, known as the "first term" or \(a_0\).
• Each number is formed by adding 2 to the previous number.
• The sequence grows by this consistent difference \(d\), which is 2.

By analyzing the sequence this way, we see a clear repeated pattern and can predict subsequent terms. Each new number is easily calculated by using these observations and trends, which simplifies understanding and computing large series terms.

Finding the Term Formula

Having identified the first term and the common difference, we can derive a formula to find any term in the sequence. This formula is known as the "term formula." It helps us find out the value of the \(n\)-th term directly without having to list all previous terms.
The formula for the \(n\)-th term \(a_n\) in an arithmetic sequence is:

• \( a_n = a_0 + n \times d \)

In our specific case:

• The first term \(a_0\) is 1.
• The common difference \(d\) is 2.

Plugging these values into the formula, we get:

• \( a_n = 1 + 2n \)

Simplifying it, the formula becomes:

• \( a_n = 2n + 1 \)

Using this formula, we can easily determine the value of any term in the sequence by substituting the term number \(n\). This makes it efficient for calculating large or specific terms without manually progressing through each term.