Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{1.8,3.6,5.4, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \( a_n = 1.8n \).

Step 1: Identify the First Term

To find the explicit formula, we first identify the first term of the sequence, which is \( a_1 = 1.8 \).

Step 2: Calculate the Common Difference

The common difference \( d \) in an arithmetic sequence is calculated by subtracting the first term from the second term. Here, \( d = 3.6 - 1.8 = 1.8 \).

Step 3: Use the Explicit Formula for Arithmetic Sequences

The explicit formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \, d \).

Step 4: Substitute Values into the Formula

Substituting the values for \( a_1 \) and \( d \) into the formula: \( a_n = 1.8 + (n-1) \times 1.8 \).

Step 5: Simplify the Formula

Simplify the formula by distributing the common difference: \( a_n = 1.8 + 1.8n - 1.8 = 1.8n \).

Key Concepts

Explicit Formula

The explicit formula is key to unraveling the pattern in arithmetic sequences. It serves as a powerful tool, allowing you to find any term in a sequence without needing to list every prior term. This formula is given by \( a_n = a_1 + (n-1) \, d \). Here, \( a_n \) represents the value of the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

For example, consider the sequence \( \{1.8, 3.6, 5.4, \ldots\} \). Here, the first term \( a_1 \) is 1.8. We can use our explicit formula to directly compute any term. This approach saves us the time and trouble of adding the common difference repeatedly until reaching the desired position.

The beauty of the explicit formula lies in its simplicity and efficiency. Once you're familiar with it, any arithmetic sequence can be tackled effortlessly.

Common Difference

The common difference in an arithmetic sequence tells us how the sequence progresses from one term to the next. It is a constant value that you can calculate by subtracting any term from the term that follows it in the series.

In the sequence \( \{1.8, 3.6, 5.4, \ldots\} \), compute the common difference \( d \) by subtracting the first term from the second: \( d = 3.6 - 1.8 = 1.8 \). This tells us that each term is obtained by adding 1.8 to the previous term.

• The common difference is crucial for forming the sequence pattern.
• It remains unchanged throughout the sequence.

Without the common difference, it would be impossible to apply the explicit formula or even extend the sequence.

Sequence Simplification

Simplifying a sequence makes it easier to understand, manage, and use. In our example, after substituting the values into the explicit formula, we get \( a_n = 1.8 + (n-1) \times 1.8 \). Simplification helps by making the formula cleaner and more user-friendly.

The process involves distributing and then reducing redundant terms. Distributing gives \( a_n = 1.8 + 1.8n - 1.8 \). Notice that the \( +1.8 \) and \( -1.8 \) terms cancel each other out, simplifying to \( a_n = 1.8n \).

A simplified sequence is:

• Easier to work with when calculating specific terms.
• More straightforward for spotting mistakes or checking work.

By reducing complexity, simplification reveals the straightforward beauty of mathematical patterns, allowing for greater focus on the sequence's core attributes.