Problem 1 - College Algebra with Modeling and Visualization

Question

Find the first four terms of the sequence. \(a_{n}=2 n+1\)

Step-by-Step Solution

Answer

The first four terms are 3, 5, 7, and 9.

Step 1: Understanding the Sequence Formula

The formula for the sequence is given by \(a_{n} = 2n + 1\). This formula allows us to find each term in the sequence by plugging in the value of \(n\), which represents the term number.

Step 2: Calculating the First Term

To find the first term \(a_1\), substitute \(n = 1\) into the sequence formula: \(a_1 = 2(1) + 1 = 2 + 1 = 3\). Thus, the first term of the sequence is 3.

Step 3: Calculating the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the sequence formula: \(a_2 = 2(2) + 1 = 4 + 1 = 5\). Thus, the second term of the sequence is 5.

Step 4: Calculating the Third Term

To find the third term \(a_3\), substitute \(n = 3\) into the sequence formula: \(a_3 = 2(3) + 1 = 6 + 1 = 7\). Thus, the third term of the sequence is 7.

Step 5: Calculating the Fourth Term

To find the fourth term \(a_4\), substitute \(n = 4\) into the sequence formula: \(a_4 = 2(4) + 1 = 8 + 1 = 9\). Thus, the fourth term of the sequence is 9.

Key Concepts

Sequence Formula

An arithmetic sequence is a list of numbers in which each term is derived from the previous one by adding a constant value, known as the common difference. Understanding the sequence formula is key to identifying the terms in an arithmetic sequence. For our given sequence, the formula is:\[a_{n} = 2n + 1\]In this equation, \(a_{n}\) represents the nth term of the sequence. The variable \(n\) stands for the position of the term in the sequence, and it must be a positive integer (1, 2, 3, and so on). By plugging different values for \(n\) into the formula, we can calculate the respective term in the sequence.This particular sequence is linear, which means that the graph of its terms forms a straight line. Understanding this can help you visualize the sequence and anticipate how it behaves as \(n\) increases.

Term Calculation

To find specific terms in the sequence, you simply substitute the desired term number into the sequence formula. Let’s see how this works step by step.

• First Term: To find \(a_1\), substitute \(n=1\): \[a_1 = 2(1) + 1 = 3\]
• Second Term: To find \(a_2\), substitute \(n=2\): \[a_2 = 2(2) + 1 = 5\]
• Third Term: To find \(a_3\), substitute \(n=3\): \[a_3 = 2(3) + 1 = 7\]
• Fourth Term: To find \(a_4\), substitute \(n=4\): \[a_4 = 2(4) + 1 = 9\]

Each term is calculated by substituting the order number of the term into \(n\). This demonstrates how consistent application of the sequence formula reveals each term reliably.

Step-by-step Solution

Following a step-by-step approach for solving sequence problems is beneficial, especially when learning the basics of sequence calculation. Here, the process is straightforward as it utilizes a simple arithmetic sequence formula.First, gather the sequence formula. In our exercise, it's already provided: \(a_{n} = 2n + 1\). Knowing this formula allows you to generate as many terms as needed by incrementing \(n\) each time.

• Step 1: Identify the formula \(a_{n} = 2n + 1\) as a sequence rule. Understand that this formula ensures every term is determined by multiplying \(n\) by 2 and adding 1.
• Step 2: Calculate each term by substituting: For \(a_1\), \(a_2\), and so on, increase \(n\) sequentially and solve for \(a_n\).
• Step 3: Verify results by rechecking your calculations. Arithmetic calculations are simple but it's easy to make little mistakes. Double-check each substitution and evaluation.

Practicing these steps will make solving such exercises quicker and increase your understanding of arithmetic sequences.