Problem 43 - College Algebra

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{3,5,7, \ldots\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = 2n + 1 \).

Step 1: Identify the First Term

In the given arithmetic sequence \( a = \{3, 5, 7, \ldots\} \), the first term \( a_1 \) is 3. This is the starting point of the sequence.

Step 2: Determine the Common Difference

To find the common difference \( d \), subtract the first term from the second term: \( d = 5 - 3 = 2 \). This signifies that each term increases by 2.

Step 3: Write the General Formula for the Arithmetic Sequence

The explicit formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \). Here, \( a_1 = 3 \) and \( d = 2 \).

Step 4: Substitute the Values into the Formula

Substitute the known values into the formula: \( a_n = 3 + (n-1) \cdot 2 \).

Step 5: Simplify the Expression

Simplify the expression to get the final explicit formula: \( a_n = 3 + 2n - 2 \). This further simplifies to \( a_n = 2n + 1 \). This is the explicit formula for the sequence.

Key Concepts

Explicit Formula

An explicit formula in the context of an arithmetic sequence is used to find any term in the sequence without requiring the preceding term. This is incredibly useful because you can jump straight to the 100th or even the 1000th term without having to manually progress through each one. The general form for the explicit formula of an arithmetic sequence is \( a_n = a_1 + (n-1) \cdot d \), where

• \( a_n \) is the term you're trying to find,
• \( a_1 \) is the first term in the sequence,
• \( n \) is the position of the term in the sequence,
• \( d \) is the common difference between consecutive terms.

To write the explicit formula, you'll first need the first term and the common difference. Then, you plug these values into the general formula and simplify. For example, if the first term is 3 and the common difference is 2, the explicit formula would be \( a_n = 3 + (n-1) \cdot 2 \). Simplifying this gives \( a_n = 2n + 1 \). This formula will allow you to calculate any term in the sequence in a straightforward manner.

Common Difference

The common difference in an arithmetic sequence is a key element that determines the nature of the sequence. It's the constant amount by which each term increases (or decreases, in the case of a negative common difference) from the previous term. In mathematical terms, if \( a_1 \) and \( a_2 \) are consecutive terms, the common difference \( d \) is calculated as \( a_2 - a_1 \).For the sequence \( \{3, 5, 7, \ldots\} \), identify the first two terms: 3 and 5. Subtract the first term from the second: \( 5 - 3 = 2 \). Therefore, the common difference \( d \) is 2.This consistent difference is what makes the sequence arithmetic, and it's pivotal when forming the explicit formula. A positive common difference indicates an increasing sequence, while a negative one indicates a decreasing sequence. Keeping the common difference in mind helps in understanding the growth pattern of the sequence, allowing for easy prediction of future terms.

First Term

The first term, often denoted as \( a_1 \), in an arithmetic sequence is the initial value that anchors the sequence. It's the starting point from which all other terms are derived using the common difference.In our given sequence, \( \{3, 5, 7, \ldots\} \), the first term \( a_1 \) is 3. Identifying this term is crucial because it helps set up the entire sequence. Together with the common difference, the first term is used to construct the explicit formula. Understanding the first term is simple: it's the number you start with in your sequence. It gives the sequence its initial value, and any modifications to this term will change the entire sequence. When given a sequence, always start by identifying the first term as it provides the basis for everything that follows.