Problem 47 - College Algebra

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 is \(a_n = 1.8n\).

Step 1: Identify the first term

The first step in writing an explicit formula for an arithmetic sequence is to identify the first term of the sequence. In the given sequence, the first term \(a_1\) is 1.8.

Step 2: Determine the common difference

Next, find the difference between any two consecutive terms to identify the common difference \(d\). In this sequence, the difference between the second term (3.6) and the first term (1.8) is \(3.6 - 1.8 = 1.8\), so \(d = 1.8\).

Step 3: Write the explicit formula

The explicit formula for the \(n\)-th term \(a_n\) of an arithmetic sequence is given by the formula \(a_n = a_1 + (n-1) \cdot d\). Substituting \(a_1 = 1.8\) and \(d = 1.8\), the formula becomes \(a_n = 1.8 + (n-1) \cdot 1.8\).

Step 4: Simplify the formula

Distribute and simplify the formula: \(a_n = 1.8 + 1.8(n-1) = 1.8 + 1.8n - 1.8 = 1.8n\). Thus, the explicit formula for the sequence is \(a_n = 1.8n\).

Key Concepts

Explicit Formula

An explicit formula is a powerful tool in mathematics that allows us to find any term in an arithmetic sequence without needing to list all preceding terms. For an arithmetic sequence, this formula takes the form \(a_n = a_1 + (n-1) \cdot d\). Here:

• \(a_n\) is the \(n\)-th term of the sequence.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference between consecutive terms.

Understanding the explicit formula lets you calculate any term directly by plugging in the term number \(n\). This formula is especially useful for large sequences because you don't have to compute every term individually to reach the one you want. For example, if you have an arithmetic sequence starting with 1.8 and a common difference of 1.8, the explicit formula becomes \(a_n = 1.8n\). This means for any term number \(n\), you simply multiply \(n\) by 1.8 to find \(a_n\).

Common Difference

In an arithmetic sequence, the common difference is a key feature that makes the sequence "arithmetic". It represents the consistent amount that separates consecutive terms in the sequence. If you know the common difference, you can navigate easily through the sequence.To find the common difference \(d\), subtract any term from the following term in the sequence. For instance, if your sequence includes terms like 1.8, 3.6, and 5.4, the difference between 3.6 and 1.8 is \(3.6 - 1.8 = 1.8\). Therefore, your common difference \(d = 1.8\).The common difference remains constant across the sequence, hence allowing for the simple calculation of each subsequent term by adding \(d\) to the previous term. This stability in difference is precisely what makes analyzing and working with arithmetic sequences straightforward.

First Term

Every arithmetic sequence begins with a specific first term, denoted as \(a_1\). This first term serves as the foundation from which the rest of the sequence is built. It is essential to know \(a_1\), as it is used in the explicit formula to calculate any term in the sequence.Let's say your sequence starts with the number 1.8, as in the case of the given sequence. This means \(a_1 = 1.8\). Knowing the first term makes it possible to establish the initial condition of your sequence, which is vital for writing the explicit formula. Without \(a_1\), you wouldn't be able to calculate subsequent terms accurately.The first term not only aids in setting up the explicit expression but also plays an integral part in helping determine the entire pattern of the sequence. It serves as the initial point from which the arithmetic progression unfolds consistently based on the common difference.