Problem 31 - Beginning and Intermediate Algebra

Question

Find the indicated term for each arithmetic sequence. $$a_{1}=-5, d=4 ; a_{16}$$

Step-by-Step Solution

Answer

The 16th term of the arithmetic sequence is \(a_{16}=55\).

Step 1: Write down the formula for finding the nth term of an arithmetic sequence

The formula for finding the nth term of an arithmetic sequence is: \(a_n = a_1 + (n-1)d\)

Step 2: Plug in the given values

We have \(a_1 = -5\), d = 4, and n = 16, so we plug these values into the formula: \(a_{16} = -5 + (16-1) \cdot 4\)

Step 3: Simplify the expression

First, we'll simplify the expression inside the parentheses and then multiply by the common difference: \(a_{16} = -5 + (15) \cdot 4\) Next, we'll multiply 15 by 4: \(a_{16} = -5 + 60\)

Step 4: Calculate the final result

Finally, we'll add -5 and 60 to get the 16th term of the arithmetic sequence: \(a_{16} = 55\) So, the 16th term of this arithmetic sequence is 55.

Key Concepts

nth term formula

The **nth term formula** is a powerful tool to find any term in an arithmetic sequence without listing all the preceding terms. This formula is given by:

• \(a_n = a_1 + (n-1)d\)

Here, \(a_n\) represents the nth term, \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference. The beauty of this formula lies in its simplicity. By substituting the given values into this formula, you can directly calculate any term in the sequence. This method saves time and enhances efficiency, especially when dealing with terms far along in the sequence.

common difference

In any arithmetic sequence, the **common difference** is a fixed number that you add to each term to get the next term. It is denoted by \(d\). For example, if your sequence starts at -5 and the common difference \(d\) is 4, each subsequent term will be 4 units larger than the previous one. Think of the common difference as the step size or the gap between consecutive terms:

• If \(d > 0\), the sequence is increasing.
• If \(d < 0\), the sequence is decreasing.
• If \(d = 0\), all terms are the same.

Recognizing and understanding the common difference enables you to predict the next terms confidently and verify the correctness of any computed terms.

sequence term calculation

**Sequence term calculation** is the process of determining a specific term in an arithmetic sequence using the parameters of the sequence. Here’s a step-by-step guide using the nth term formula:
1. Identify \(a_1\) and \(d\), the first term and common difference of the sequence.
2. Choose the term number \(n\) you want to find.
3. Substitute these values into the nth term formula: \(a_n = a_1 + (n-1)d\).
4. Simplify the expression to find the desired term.
This procedural approach ensures accuracy and allows you to solve even complex problems without confusion.

finding terms in a sequence

**Finding terms in a sequence** means understanding the pattern and using it to determine specific terms. Using the nth term formula is often the most straightforward method.To find a term:

• Start with the first term \(a_1\) of the sequence.
• Identify the common difference \(d\).
• Plug these values into \(a_n = a_1 + (n-1)d\).
• Perform the arithmetic calculations.

In the case of the original exercise, knowing that \(a_1 = -5\), \(d = 4\), allows you to find the 16th term effortlessly. Whether for homework or exams, mastering this skill offers valuable insight into the structure and behaviors of arithmetic sequences.