Problem 23 - Beginning and Intermediate Algebra

Question

Given the arithmetic sequence \(4,-1,-6,-11,-16, \dots\) a) Find \(a_{1}\) and \(d\) b) Find a formula for the general term of the sequence, \(a_{n}\) c) Find \(a_{19}\)

Step-by-Step Solution

Answer

a) \(a_1=4\), \(d=-5\) b) \(a_n = 4 + (n-1)(-5)\) c) \(a_{19} = -86\)

Step 1: Find \(a_{1}\) and \(d\)

Since we are given the sequence \(4,-1,-6,-11,-16, \dots\), it is easy to see that the first term of the sequence is \(a_1 = 4\). The common difference, \(d\), can be found by subtracting the first term from the second term, or any other consecutive terms. In this case, we have \(-1 - 4 = -5\). So, \(d = -5\).

Step 2: Find the formula for the general term, \(a_n\)

To find the general term of the arithmetic sequence, we use the formula: \[a_n = a_1 + (n-1)d\] In our case, \(a_1 = 4\) and \(d = -5\). Substituting these values into the formula gives: \[a_n = 4 + (n-1)(-5)\]

Step 3: Find \(a_{19}\)

Now that we have the general term formula, we can find the 19th term, \(a_{19}\), by substituting \(n=19\) into the formula: \[a_{19} = 4 + (19-1)(-5)\] Calculate the value inside the parenthesis first: \[(19-1) = 18\] Now, multiply 18 by -5: \[18 \times (-5) = -90\] Finally, add 4 to -90: \[a_{19} = 4 - 90 = -86\] So, the 19th term of the given arithmetic sequence is \(a_{19} = -86\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the common difference is a key feature that helps define the sequence. It is the constant amount you add or subtract to get from one term to the next. In our given sequence, the terms are 4, -1, -6, -11, and so on. To find the common difference, look at any two consecutive terms and subtract the earlier term from the later one. For instance, subtracting the first term, 4, from the second term, -1, we find

• \(-1 - 4 = -5\).

This means our common difference, \(d\), is -5.
By regularly adding -5 to each term, a pattern emerges, allowing us to predict every subsequent term in the sequence. It's essential to identify the common difference clearly because it dictates how the sequence progresses. Once found, this value can be used to form the general term of the sequence, making it much easier to find specific terms later on.

Deriving the General Term of a Sequence

The general term formula in an arithmetic sequence is a powerful tool for describing the position of any term within that sequence. It is given by the formula:

• \(a_n = a_1 + (n-1)d\),

where \(a_n\) represents the term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
For our sequence, \(a_1 = 4\) and \(d = -5\). Plugging these values into the general formula, we can calculate the nth term as

• \(a_n = 4 + (n-1)(-5)\).

This formula simplifies calculating any term directly by simply substituting the value of \(n\). The flexibility and ease of use make this formula a fundamental part of working with arithmetic sequences. Once this formula is in place, it opens the door to find specific terms without recalculating step-by-step.

Finding Specific Terms in an Arithmetic Sequence

Once the general term formula is established, you can easily find any term you need in the arithmetic sequence. To illustrate this, consider finding the 19th term in our sequence. Using the general formula, replace \(n\) with 19 to find:

• \(a_{19} = 4 + (19-1)(-5)\).

Start by calculating inside the parentheses:

• \(19-1 = 18\).

Then, multiply this by the common difference:

• \(18 \times (-5) = -90\).

Finally, add the first term to this product:

• \(4 + (-90) = -86\).

Hence, the 19th term is \(a_{19} = -86\).
This straightforward plug-and-chug procedure makes it efficient to find any specific term in the series, regardless of its position, without listing all preceding terms. Understanding the use of the general term formula transforms the process, making it manageable and clear.