Problem 44 - College Algebra

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{32,24,16, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = 40 - 8n \).

Step 1: Identify the First Term and Common Difference

An arithmetic sequence is defined by its first term and common difference. The first term here is \( a_1 = 32 \). To find the common difference \( d \), subtract the first term from the second term: \( 24 - 32 = -8 \). Therefore, the common difference \( d = -8 \).

Step 2: Write the General Formula for Arithmetic Sequence

The explicit formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \). Substitute \( a_1 = 32 \) and \( d = -8 \) into this formula.

Step 3: Simplify the Formula

Substitute into the general formula: \( a_n = 32 + (n-1)(-8) \). Simplify the equation: \( a_n = 32 - 8n + 8 \). Thus, \( a_n = 40 - 8n \) is the explicit formula for the sequence.

Key Concepts

Explicit Formula

The explicit formula is a mathematical equation that allows us to find any term in an arithmetic sequence without calculating the previous terms. This saves time, especially when working with sequences having many terms. To derive an explicit formula, you need to know two crucial components: the first term \(a_1\) and the common difference \(d\). The formula for an arithmetic sequence is:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, \(a_n\) represents the term we want to find, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. By substituting the known values for \(a_1\) and \(d\), we can directly calculate any term in the sequence. In our example, substituting gives us \(a_n = 40 - 8n\). This formula lets you easily determine any term by plugging in the value of \(n\).

Common Difference

The common difference in an arithmetic sequence is the value you add (or subtract) to each term to get to the next one. It plays a key role in defining the pattern of the sequence. To find the common difference \(d\), subtract the first term from the second term.

• Example calculation: If the sequence starts as 32, 24, 16, the common difference is \(24 - 32 = -8\).

A negative common difference like \(-8\) indicates that each term is decreasing by 8 as you progress through the sequence. Recognizing this consistent change not only helps in finding the explicit formula but also aids in understanding the direction in which the sequence is moving—whether it’s increasing or decreasing. This understanding is crucial for predicting how sequences will behave further along.

First Term

The first term in an arithmetic sequence is the initial value from which the sequence begins. It is denoted by \(a_1\). In arithmetic sequences, the first term is critical because it sets the starting point for the entire sequence. Without it, you cannot correctly build the sequence or derive the explicit formula. In our specific example, \(a_1 = 32\) is the first term. This means our sequence begins at 32, and subsequent terms are determined by applying the common difference repeatedly to this starting value. The clear identification of the first term, along with the common difference, is essential in accurately representing the entire sequence both in list form and when writing formulas. By knowing \(a_1\), one can easily generate further terms in conjunction with the explicit formula—each step leads naturally into the next as dictated by the arithmetic pattern.