Problem 16 - College Algebra

Question

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is \(5,\) common difference is \(6,\) fi \(d\) the \(8^{\text {th }}\) term.

Step-by-Step Solution

Answer

The 8th term of the sequence is 47.

Step 1: Understand the problem

We are given the first term of an arithmetic sequence, denoted as \(a_1 = 5\), and the common difference \(d = 6\). We are asked to find the 8th term of this sequence.

Step 2: Recall the formula for the nth term of an arithmetic sequence

The formula to find the \(n^{th}\) term of an arithmetic sequence is given by: \(a_n = a_1 + (n-1) \times d\). We will use this formula to find the 8th term.

Step 3: Substitute the known values into the formula

Substitute \(a_1 = 5\), \(d = 6\), and \(n = 8\) into the formula: \(a_n = 5 + (8-1) \times 6\).

Step 4: Simplify the expression

Work through the calculation: \((8-1) = 7\) and then \(7 \times 6 = 42\). Therefore, \(a_8 = 5 + 42\).

Step 5: Calculate the 8th term

Finally, add the numbers: \(5 + 42 = 47\). Therefore, the 8th term of the sequence is \(47\).

Key Concepts

Understanding the Common Difference

In arithmetic sequences, the **common difference** is the number that you add (or subtract, if it's negative) to move from one term to the next. This difference remains constant throughout the sequence. For example, if you have the sequence 2, 5, 8, 11,..., the common difference here is 3 because each term increases by 3.
To find the common difference:

• Subtract any term from the term that follows it.
• Ensure the difference is consistent between all successive terms.

In our exercise, we specifically start with the term 5, and each subsequent term is 6 more than the previous one. Therefore, the common difference is 6. Understanding this key concept allows you to predict how the sequence will behave and helps in accurately calculating any term in the sequence.

Using the nth Term Formula

Knowing how to use the **nth term formula** is the heart of working with arithmetic sequences. This formula helps us determine any term within the sequence without listing all previous terms. The formula is:
\[ a_n = a_1 + (n-1) \times d \]
Where:

• \(a_n\) is the nth term you are looking for.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the term number in the sequence.

The **nth term formula** streamlines finding specific terms by allowing direct substitution of these values. For example, to find the 8th term in our exercise, you quickly substitute 5 for \(a_1\), 6 for \(d\), and 8 for \(n\) in the formula, yielding the simplified expression. The result is straightforward once you plug in these values.

Performing Sequence Calculation

Once you understand the arithmetic sequence's structure and have used the nth term formula, you can smoothly execute the **sequence calculation** to find the desired term. It's essentially plugging numbers into a formula:

First, compute inside the parentheses: \((n-1)\). In our example, \((8-1) = 7\).
Next, multiply by the common difference:\(7 \times 6 = 42\).
Finally, add this product to the first term: \(5 + 42 = 47\).
These steps reflect the logical succession in arithmetic sequence calculations. Each step builds on the last, making them intuitive and straightforward once practiced a few times. It's this systematic approach that ensures accuracy and efficacy in finding any sequence term.