Problem 21 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=x, a_{2}=x+3$$

Step-by-Step Solution

Answer

\(a_8 = x + 21\) and \(a_n = x + 3n - 3\).

Step 1: Identify the First Term

The initial term of the arithmetic sequence, denoted as \(a_1\), is given as \(x\). This is our starting point for constructing the sequence.

Step 2: Find the Common Difference

To find the common difference \(d\) of the arithmetic sequence, subtract the first term from the second term: \(d = a_2 - a_1 = (x+3) - x = 3\).

Step 3: Use the General Formula to Find \(a_8\)

The general formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \times d\). To find \(a_8\), let \(n = 8\): \(a_8 = x + (8-1) \times 3 = x + 21\).

Step 4: Use the General Formula to Find \(a_n\)

Apply the general formula for the \(n\)-th term: \(a_n = a_1 + (n-1) \times d = x + (n-1) \times 3\), which simplifies to \(a_n = x + 3n - 3\).

Key Concepts

Common Difference

In an arithmetic sequence, every term is created by adding a fixed number, called the common difference, to the previous term. This difference helps determine the sequence's progression and is crucial for identifying the nature of the sequence.
The common difference, often denoted as \(d\), is calculated by subtracting the first term from the second term. For example, if the first term \(a_1\) of a sequence is \(x\) and the second term \(a_2\) is \(x + 3\), then the common difference \(d\) is:

• \(d = a_2 - a_1 = (x + 3) - x = 3\)

This value of 3 tells us that each term is 3 greater than the previous one. Understanding the common difference ensures that you can predict the sequence's pattern accurately.
Once you know this consistent increment, you can easily use it to determine any term in the sequence by applying it repeatedly to preceding terms.

General Formula

The general formula for any arithmetic sequence allows you to find any term based on its position in the sequence. This formula is essential for predicting or finding specific terms without having to list all preceding ones.
In an arithmetic sequence, the general formula for the \(n\)-th term \(a_n\) is given by:

• \(a_n = a_1 + (n-1) \times d\)

Here, \(a_1\) is the first term of the sequence, \(d\) is the common difference, and \(n\) is the term number you wish to find.
For example, suppose the first term \(a_1 = x\) and the common difference \(d = 3\), then:

• The general term becomes \(a_n = x + (n-1) \times 3\)
• This formula simplifies to \(a_n = x + 3n - 3\)

This gives a clear method to calculate any term \(a_n\) knowing its position \(n\) while harnessing the consistent addition resulting from \(d\).
The general formula is particularly useful when dealing with large sequences and looking for specific terms far into it.

Nth Term

The \(n\)-th term in an arithmetic sequence is the term that is located at the position \(n\). By using the general formula, you can accurately find this term irrespective of how far it is from the first term. Finding the \(n\)-th term boils down to substituting \(n\) into the arithmetic formula, which is often:

• \(a_n = a_1 + (n-1) \times d\)

Taking the given exercise where \(a_1 = x\) and \(d = 3\), for the \(n\)-th term, we substitute and simplify:

• \(a_n = x + (n-1) \times 3\)
• Simplified, this becomes \(a_n = x + 3n - 3\)

This allows you to replace \(n\) with any number to find the term at that particular position.
For instance, if you need to find the 8th term, plug in 8 for \(n\):
\(a_8 = x + 3 \times 8 - 3 = x + 24 - 3 = x + 21\).
The \(n\)-th term formula is therefore a straightforward way to work out terms beyond the first few without performing repeated additions.