Problem 105 - College Algebra

Question

Use the formula \(a_{n}-4+(n-1)(-7)\) to find the eighth term of the sequence \(4,-3,-10, \ldots\)

Step-by-Step Solution

Answer

The eighth term of the sequence is -45.

Step 1: Understand the formula

An arithmetic sequence is given by \(a_{n} = a_{1} + (n-1)d\), where \(a_{1}\) is the first term, n is the term number, and d is the common difference between terms. In this case, the formula given is \(a_{n} = 4 + (n-1)(-7)\). This means that the first term \(a_{1}\) is 4 and the common difference d is -7.

Step 2: Plug in the relevant values

We are asked to find the eighth term of the sequence, so we plug in n = 8 into the formula: \(a_{8} = 4 + (8-1)(-7)\).

Step 3: Simplify the expression

Next, execute the operations in the parenthesis and the multiplication: \(a_{8} = 4 + 7*(-7) = 4 - 49\).

Step 4: Final Calculation

Finally, subtract 49 from 4 to obtain the eighth term: \(a_{8} = 4 - 49 = -45\).

Key Concepts

Common Difference

When dealing with an arithmetic sequence, understanding the **common difference** is crucial. The common difference is the consistent amount added to each term to arrive at the next term. It's an essential part of identifying the pattern within the sequence.

In our sequence, the common difference (\(d\)) is -7. This means that starting from the first term, each subsequent term is determined by subtracting 7. For example:

• The first term (\(a_1\)) is 4.
• The second term is calculated as 4 - 7 = -3.
• The third term is -3 - 7 = -10, and so forth.

Recognizing that the common difference is negative (-7) helps us understand why each term in the sequence decreases. This pattern is vital for calculating any specific term within the sequence.

Term Number

The **term number** represents the position of a specific term within the sequence. It is denoted as \(n\). For any arithmetic sequence, the term number tells us which position we are interested in finding or evaluating.

In this particular exercise, we were asked to find the eighth term, which means \(n = 8\). Knowing the term number helps us plug the correct value into the arithmetic formula to find our desired term.

In practice:

• Term 1: We start with the first term, which is 4.
• Term 2: Next is -3, achieved by applying the common difference.
• Continue this process to reach the eighth term.

This systematic approach ensures that we accurately apply the sequence's rules to find any term.

Sequence Formula

The **sequence formula** in arithmetic sequences is a powerful tool that enables us to compute any term directly, without listing all previous terms. It is expressed generally as:\[a_n = a_1 + (n-1) \, d\]

This formula, tailored for our specific sequence as \(a_n = 4 + (n-1)(-7)\), contains several key components:

• \(a_1\): the first term of the sequence, which here is 4.
• \(n\): the term number you are interested in obtaining.
• \(d\): the common difference, calculated as -7 for this sequence.

Using this formula, we can determine any term by substituting \(n\) with the term number we are interested in. For instance, to find the eighth term, we set \(n = 8\) in the formula and calculate accordingly. This method saves time and steps, especially for large term numbers, making it a highly efficient approach in arithmetic sequences.