Problem 21 - College Algebra

Question

For the following exercises, fi d the fi st term given two terms from an arithmetic sequence. Find the fi st term or \(a_{1}\) of an arithmetic sequence if \(a_{8}=40\) and \(a_{23}=115\).

Step-by-Step Solution

Answer

The first term, \(a_1\), of the arithmetic sequence is 5.

Step 1: Identify the common difference

In an arithmetic sequence, the formula for the nth term is given by: \(a_n = a_1 + (n-1) imes d\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference. We know \(a_8 = 40\) and \(a_{23} = 115\). Let's set up two equations using the given terms: 1. \(a_8 = a_1 + 7d = 40\) 2. \(a_{23} = a_1 + 22d = 115\). Subtract the first equation from the second to find the common difference \(d\).

Step 2: Solve for the common difference

Subtract the two equations:\((a_1 + 22d) - (a_1 + 7d) = 115 - 40\)which simplifies to:\(15d = 75\). Divide both sides by 15 to solve for \(d\): \(d = 5\). This is the common difference.

Step 3: Use the common difference to find the first term

Substitute \(d = 5\) back into the equation for \(a_8\): \(a_8 = a_1 + 7 imes 5 = 40\). Simplify: \(a_1 + 35 = 40\). Subtract 35 from both sides to find \(a_1\): \(a_1 = 5\).

Key Concepts

Understanding the First Term

In arithmetic sequences, the first term is the starting point of the sequence. It is known as \(a_1\), and it sets the stage for the entire sequence by establishing its base value. Understanding this first term is crucial because every other term builds upon it through the addition of the common difference.
For example, if you know that the first term \(a_1 = 5\), you have your starting point. Each subsequent term will add the common difference to this value to find the next term, continuing the pattern. Recognizing the first term helps to reconstruct the sequence from any point, given enough pieces of information like other terms or the common difference.

Exploring the Common Difference

The common difference in an arithmetic sequence is a constant that is added to each term to get the next term. It plays a vital role in determining the pattern and progression of the sequence.
Mathematically, the common difference \(d\) can be found when you know at least two terms in the sequence. Suppose you have \(a_8 = 40\) and \(a_{23} = 115\). By setting up equations based on these terms:

• \(a_8 = a_1 + 7d = 40\)
• \(a_{23} = a_1 + 22d = 115\)

you can subtract the first equation from the second, which results in: \(15d = 75\). Solving this equation gives \(d = 5\). This common difference allows each term in the sequence to be uniquely defined from its predecessors.

The Power of the nth Term Formula

The nth term formula is a crucial tool in arithmetic sequences. It allows you to find any term in the sequence using just the first term \(a_1\) and the common difference \(d\). The formula is written as \(a_n = a_1 + (n-1) \times d\).
This formula is powerful because it enables quick calculations without the need to manually add the common difference repeatedly. For instance, to find \(a_8\), we use the formula:

• \(a_8 = a_1 + 7 \times d = 5 + 7 \times 5 = 40\)

Hence, you can directly calculate the eighth term using the formula if you have \(a_1 = 5\) and \(d = 5\). By using this formula, any term can be precisely pinpointed, making it an essential tool in arithmetic sequences.