Problem 48 - College Algebra

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-18.1,-16.2,-14.3, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \(a_n = 1.9n - 20.0\).

Step 1: Identify the First Term (a_1)

The first term of the sequence given is \(-18.1\). This will be our \(a_1\) in the explicit formula.

Step 2: Determine the Common Difference (d)

To find the common difference \(d\), subtract the first term from the second term: \(-16.2 - (-18.1) = 1.9\). The common difference \(d\) is \(1.9\).

Step 3: Write the Explicit Formula

The explicit formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \, d \). Plug in the values: \( a_n = -18.1 + (n-1) \, 1.9 \).

Step 4: Simplify the Formula

Distribute the common difference and simplify: \( a_n = -18.1 + 1.9n - 1.9 \). Combine like terms to get: \( a_n = 1.9n - 20.0 \). This is the explicit formula for the sequence.

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression that allows us to find any term in a sequence quickly, without needing to know the previous terms. This is particularly useful in arithmetic sequences where the difference between consecutive terms is constant. In an arithmetic sequence, the explicit formula is expressed as:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, \( a_n \) represents the nth term of the sequence, \( a_1 \) is the first term, and \( d \) is the common difference. This formula helps us determine the value of any term directly by plugging in the desired term's position \( n \).
For example, if you want to find the 10th term of the sequence \(-18.1, -16.2, -14.3, \ldots\), just replace \( n \) with 10 in the explicit formula.

Common Difference

The common difference in an arithmetic sequence is the fixed amount added to each term to get the next term. It is a crucial part of the sequence's structure because it defines how the sequence progresses. The common difference is denoted by \( d \) and can be found by subtracting any term in the sequence from the term immediately following it.
In our example, the sequence is \(-18.1, -16.2, -14.3, \ldots\). By subtracting the first term \(-18.1\) from the second term \(-16.2\), you get the common difference:

• \(-16.2 - (-18.1) = 1.9\)

With a common difference of \(1.9\), you know that each term is 1.9 units greater than the previous one.
This information is essential for writing the explicit formula and understanding the sequence's behavior.

First Term

The first term of an arithmetic sequence, denoted as \( a_1 \), is the initial value from which the sequence starts. Knowing the first term is fundamental because it serves as the starting point for developing the explicit formula.
In our given sequence \(-18.1, -16.2, -14.3, \ldots\), the first term is \(-18.1\). This means that all subsequent terms are generated beginning with this starting value, increasing by the common difference each time.
When writing the explicit formula, the first term \( a_1 \) is essential, as it is plugged directly into the formula to allow you to calculate any subsequent term. Without the first term, the explicit formula would not work because it would lack a starting point.