Problem 26 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) for each arithmetic sequence. $$a_{12}=60, a_{20}=84$$

Step-by-Step Solution

Answer

The first term, \(a_1\), is 27.

Step 1: Understand the Arithmetic Sequence Formula

In an arithmetic sequence, each term is the sum of the previous term and a constant difference. The general formula for the nth term is \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference between terms.

Step 2: Write Equations Using Known Terms

We know two terms of the sequence: \(a_{12} = 60\) and \(a_{20} = 84\). We'll write these as equations:1. \(a_{12} = a_1 + 11d = 60\)2. \(a_{20} = a_1 + 19d = 84\)These equations will help us solve for \(a_1\) and \(d\).

Step 3: Subtract the Two Equations

Subtract the first equation from the second to eliminate \(a_1\):\((a_1 + 19d) - (a_1 + 11d) = 84 - 60\)This simplifies to:\(19d - 11d = 24\)Or:\(8d = 24\).

Step 4: Solve for the Common Difference \(d\)

Divide both sides of the equation by 8:\(d = \frac{24}{8} = 3\).

Step 5: Substitute \(d\) Back Into One Equation

Use the value of \(d = 3\) in one of the original equations to solve for \(a_1\). Using \(a_1 + 11d = 60\):\(a_1 + 11(3) = 60\)\(a_1 + 33 = 60\).

Step 6: Solve for the First Term \(a_1\)

Subtract 33 from both sides to solve for \(a_1\):\(a_1 = 60 - 33\)\(a_1 = 27\).

Key Concepts

nth term formula

In the realm of arithmetic sequences, understanding the formula for the nth term is crucial. The formula allows us to find any term in a sequence using its position number, represented by \( n \). The formula is given by:

• \( a_n = a_1 + (n - 1)d \)

Here, \( a_n \) is the term of interest, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference between each term.
Let's break it down: each term in the sequence is formed by adding the common difference \( d \) repeatedly, starting from the first term. The formula uses \( (n-1) \) because the first term doesn't need any additions of \( d \). It simply sets the stage for all subsequent terms. This formula is essential because, with it, you can calculate any term in the sequence quickly, as long as you know \( a_1 \) and \( d \).

common difference

A key feature of arithmetic sequences is the "common difference." This is the value, denoted by \( d \), that is added to each term to get to the next one. The common difference is what makes each arithmetic sequence distinctive and predictable. To find the common difference:

• Identify any two consecutive terms.
• Subtract the smaller term from the larger one.
• The result is your \( d \).

For instance, if you have a sequence where \( a_{12} = 60 \) and \( a_{20} = 84 \), you can use these two terms to determine that \( 8d = 24 \). Solving gives \( d = 3 \).
This means each term in the sequence increases by 3 units. Understanding \( d \) allows you to construct the entire sequence step-by-step from any starting point.

first term of sequence

The first term, denoted as \( a_1 \), is the starting point of any arithmetic sequence. It's the initial building block from which all subsequent terms are derived through the addition of the common difference. When solving arithmetic sequence problems, one often needs to find \( a_1 \) based on other known terms and the common difference.

• Use the nth term formula: \( a_n = a_1 + (n - 1)d \).
• Substitute known values of \( a_n \), \( d \), and \( n \) into the formula.
• Solve the equation for \( a_1 \).

In our example, we found that \( d = 3 \). By using this value in the equation \( a_{12} = 60 \), we are able to mathematically manipulate it to find \( a_1 = 27 \). Solving for the first term provides the foundation for creating the entire sequence, and knowing \( a_1 \) along with \( d \) allows the calculation of any term within the sequence efficiently.