Problem 4 - College Algebra with Modeling and Visualization

Question

Find the first four terms of the sequence. \(a_{n}=2(3)^{n}\)

Step-by-Step Solution

Answer

The first four terms are 6, 18, 54, and 162.

Step 1: Understand the Problem

We are asked to find the first four terms of the sequence defined by the formula \(a_n = 2(3)^n\). This means we need to substitute \(n = 1, 2, 3,\) and \(4\) into the formula and calculate the corresponding terms.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the sequence formula: \(a_1 = 2(3)^1 = 2 \times 3 = 6\). The first term is 6.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the sequence formula: \(a_2 = 2(3)^2 = 2 \times 9 = 18\). The second term is 18.

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the sequence formula: \(a_3 = 2(3)^3 = 2 \times 27 = 54\). The third term is 54.

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the sequence formula: \(a_4 = 2(3)^4 = 2 \times 81 = 162\). The fourth term is 162.

Step 6: Summarize the Terms

The first four terms of the sequence are 6, 18, 54, and 162.

Key Concepts

Sequence Formula

A sequence formula is a mathematical equation that represents the relationship between the position of a term in a sequence and its value. In simple terms, it's like a recipe that tells you how to calculate each term of the sequence based on its position. In the example we are dealing with, the sequence is defined by the formula: \(a_n = 2(3)^n\), where \(a_n\) represents the sequence terms, and \(n\) is the position of the term in the sequence. This formula is particularly for a geometric sequence, which means each term is derived by multiplying the previous term by a constant. To solve problems like this, identify your sequence formula first. The formula will provide a blueprint to extract the terms needed. Understand what each part of the formula signifies and how changes in \(n\) affect the results.

Term Calculation

Term calculation involves using the sequence formula to determine the exact value of each term in the sequence. Let's break down the steps to calculate the terms for clarity. First, substitute the specific term number \(n\) into the formula \(a_n = 2(3)^n\). This substitution will give you a new equation for every term position \(n = 1, 2, 3,\) and \(4\). For example:

• Substitute \(n = 1\) to find the first term: \(a_1 = 2(3)^1 = 6\).
• Substitute \(n = 2\) for the second term: \(a_2 = 2(3)^2 = 18\).
• Substitute \(n = 3\) to find the third term: \(a_3 = 2(3)^3 = 54\).
• Finally, substitute \(n = 4\) to get the fourth term: \(a_4 = 2(3)^4 = 162\).

These calculations illustrate how each term is calculated individually by plugging numbers into the sequence formula. Knowing how to do this will enable you to find any term in the sequence.

Pattern Recognition

Recognizing patterns in sequences helps in understanding their behavior and predicting future terms. For geometric sequences like the one we're analyzing, pattern recognition revolves around identifying the common ratio between terms.In this case, our sequence \(a_n = 6, 18, 54, 162\) displays a consistent pattern where each term is multiplied by the same factor, known as the common ratio. Here, the common ratio is 3, meaning each term is 3 times the previous term.Pattern recognition aids in predicting terms without recalculating everything. Simply take the last known term and multiply it by the common ratio. This skill becomes especially useful when needing to find terms in more complex sequences or verifying your calculations efficiently. Noting such patterns helps streamline the process and deepen comprehension of the sequence's structure.