Problem 44 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{4}=12, d=-10$$f

Step-by-Step Solution

Answer

The general term is \( a_n = 52 - 10n \).

Step 1: Understanding the Problem

We are given an arithmetic sequence with the 4th term, \( a_4 = 12 \), and a common difference, \( d = -10 \). We need to find the general formula for the \( n \)-th term, \( a_n \).

Step 2: General Formula for Arithmetic Sequence

The general formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step 3: Using Provided Information

We have \( a_4 = 12 \) and \( d = -10 \). We can use the general formula for the 4th term: \( a_4 = a_1 + (4-1)(-10) = a_1 - 30 \). Hence, \( a_1 - 30 = 12 \).

Step 4: Solving for the First Term

By rearranging \( a_1 - 30 = 12 \), we find \( a_1 = 12 + 30 = 42 \). Therefore, the first term \( a_1 = 42 \).

Step 5: Writing the General Term

Now that we know \( a_1 = 42 \) and \( d = -10 \), we can substitute these into the general formula: \( a_n = 42 + (n-1)(-10) \). Simplified, this is \( a_n = 42 - 10(n-1) \).

Step 6: Simplifying the General Term

Simplifying the expression \( a_n = 42 - 10(n-1) \), we get: \( a_n = 42 - 10n + 10 \) which further simplifies to \( a_n = 52 - 10n \).

Key Concepts

General Formula

In arithmetic sequences, the **general formula** is a critical concept that helps us find any term in the sequence. An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant value, known as the **common difference**, to the previous term. The formula for the nth term in an arithmetic sequence is given by:\[ a_n = a_1 + (n-1)d \]Where:

• \( a_n \) is the nth term that we want to find.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between the terms.
• \( n \) is the position of the term in the sequence.

This formula is powerful because it allows you to generate terms without having to list all preceding terms. It gives insight into the structure and pattern of the sequence.

Common Difference

The **common difference** is the backbone of an arithmetic sequence. It is the fixed amount added to each term to get the next one in the sequence. In the given problem, the common difference, \( d \), is \(-10\).Understanding this element is crucial. Why?

• It determines how quickly the sequence grows or shrinks.
• This value is what differentiates an arithmetic sequence from other types of sequences.

The sign of \( d \) affects the progression:

• If \( d > 0 \), the sequence increases.
• If \( d < 0 \), as in our example, the sequence decreases.

In practical terms, knowing \( d \) allows you to predict future values and understand the pattern the sequence follows.

First Term

The **first term** of an arithmetic sequence, denoted as \( a_1 \), is where everything begins. It's the anchor that holds the sequence in place. For our problem, after calculation, we know \( a_1 = 42 \).Why is the first term important?

• It serves as the starting point for generating all other terms in the sequence.
• Knowing \( a_1 \) allows you to use the general formula to calculate any term without errors.
• In some cases, it might provide context or initial condition for a real-world scenario modeled by the sequence.

Even though the first term often seems like just another number, it fundamentally defines the sequence along with the common difference.