Problem 18 - Beginning and Intermediate Algebra

Question

Write the first five terms of the arithmetic sequence with general term \(a_{n}\). $$a_{n}=2 n+7$$

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence with general term \(a_n = 2n + 7\) are: \(a_1 = 9, \; a_2 = 11, \; a_3 = 13, \; a_4 = 15, \; a_5 = 17\).

Step 1: Understand the given formula

The given formula represents the general term of the arithmetic sequence. For this sequence, the formula is \(a_n = 2n + 7\), where \(n\) is the position of the term in the sequence, and \(a_n\) is the term's value.

Step 2: Find the first term, \(a_1\)

To find the first term, replace \(n\) with 1 in the general term formula, \(a_n = 2n + 7\). Thus, \(a_1 = 2(1) + 7 = 2 + 7 = 9\).

Step 3: Find the second term, \(a_2\)

To find the second term, replace \(n\) with 2 in the general term formula, \(a_n = 2n + 7\). Thus, \(a_2 = 2(2) + 7 = 4 + 7 = 11\).

Step 4: Find the third term, \(a_3\)

To find the third term, replace \(n\) with 3 in the general term formula, \(a_n = 2n + 7\). Thus, \(a_3 = 2(3) + 7 = 6 + 7 = 13\).

Step 5: Find the fourth term, \(a_4\)

To find the fourth term, replace \(n\) with 4 in the general term formula, \(a_n = 2n + 7\). Thus, \(a_4 = 2(4) + 7 = 8 + 7 = 15\).

Step 6: Find the fifth term, \(a_5\)

To find the fifth term, replace \(n\) with 5 in the general term formula, \(a_n = 2n + 7\). Thus, \(a_5 = 2(5) + 7 = 10 + 7 = 17\).

Step 7: Write the first five terms of the sequence

Now that we've found the first five terms of the arithmetic sequence, we can write them out as follows: $$a_1 = 9, \quad a_2 = 11, \quad a_3 = 13, \quad a_4 = 15, \quad a_5 = 17$$

Key Concepts

General Term Formula

The general term formula is the foundation of any arithmetic sequence. It allows us to calculate any term of the sequence without listing all the preceding ones. In this problem, the general term formula given is \(a_n = 2n + 7\). This formula indicates how to find the value of the sequence’s nth term:

• \(n\) denotes the position of the term in the sequence.
• \(a_n\) is the actual value of the term at position \(n\).
• The expression \(2n + 7\) defines the pattern or rule of the sequence.

The formula reveals that for each successive term, we increase the position by 1 and multiply by 2, then add 7. This pattern enables us to quickly find any term, making the sequence predictable and easy to work with.

First Five Terms

Finding the first five terms involves substituting values into the general term formula. Let's walk through this step-by-step:

• First term \(a_1\): Substitute \(n = 1\) into \(a_n = 2n + 7\), we get \(a_1 = 2(1) + 7 = 9\).
• Second term \(a_2\): Now substitute \(n = 2\), leading to \(a_2 = 2(2) + 7 = 11\).
• Third term \(a_3\): Substitute \(n = 3\), giving \(a_3 = 2(3) + 7 = 13\).
• Fourth term \(a_4\): Substitute \(n = 4\), resulting in \(a_4 = 2(4) + 7 = 15\).
• Fifth term \(a_5\): Finally, substitute \(n = 5\) to find \(a_5 = 2(5) + 7 = 17\).

The calculated terms show how the values increase as we move along the sequence: 9, 11, 13, 15, and 17. This sequence showcases a constant difference, known as the common difference, which helps reinforce the arithmetic properties.

Step-by-Step Solution

A step-by-step solution provides a clear and thorough understanding of how each term in the sequence is derived. It breaks down the task into manageable stages, assisting learners in connecting the theoretical formula to its practical application:

1. Understand the given formula: Identify what each part of the formula refers to and the sequence rule it represents.
2. Replace \(n\) with the term’s position: For each of the first five terms (\(n = 1\) through \(n = 5\)), substitute into the formula to determine the term value.
3. Calculate the algebraic expression: Perform the necessary arithmetic operations—multiply and then add—to compute the exact value.
4. Document the results: After calculating, list the terms in order to visualize the sequence.
Each step builds onto the next, ensuring comprehensive understanding. This also promotes problem-solving skills and confidence in tackling similar exercises independently.