Problem 27 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) for each arithmetic sequence. $$a_{5}=-3, a_{18}=-29$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant difference is known as the common difference, denoted by \(d\).

Step 2: Use the Formula for the nth Term

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. We have two specific terms: \(a_5 = -3\) and \(a_{18} = -29\).

Step 3: Set Up Equations for Known Terms

Use the formula to express the given terms in terms of \(a_1\) and \(d\):1. For \(a_5\): \(-3 = a_1 + 4d\)2. For \(a_{18}\): \(-29 = a_1 + 17d\).

Step 4: Solve the System of Equations

To find \(a_1\) and \(d\), we'll solve the two equations simultaneously. Subtract the first equation from the second:\[-29 = a_1 + 17d\]\[-3 = a_1 + 4d\]Subtract: \[-26 = 13d\]Thus, \(d = -2\).

Step 5: Find the First Term \(a_1\)

Substitute \(d = -2\) into the equation for \(a_5\):\[-3 = a_1 + 4(-2)\]Simplify: \[-3 = a_1 - 8\]Solve for \(a_1\): \[a_1 = -3 + 8 = 5\].

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the constant amount that each term differs from the previous one. It allows us to form the structure of the sequence. In our example, we have the expressions for the 5th and 18th terms of the sequence:

• Equation for the 5th Term: \( a_5 = a_1 + 4d \)
• Equation for the 18th Term: \( a_{18} = a_1 + 17d \)

The common difference, denoted as \(d\), is found by solving these equations. By subtracting the equation for \(a_5\) from that of \(a_{18}\), we isolate \(d\). This concept is core because it quantifies the rate of change in an arithmetic sequence, letting us accurately describe every term in the sequence.

Equation Solving

Equation solving involves finding unknown values by manipulating mathematical expressions. In our case, finding the first term \(a_1\) and the common difference \(d\) requires us to solve a system of linear equations. Starting with:

• \( -3 = a_1 + 4d \)
• \( -29 = a_1 + 17d \)

We subtract the first equation from the second, which eliminates \(a_1\), allowing us to solve for \(d\). This approach highlights balancing and reducing equations to solve for particular variables – a fundamental skill in algebra and many applications. Once we know \(d\), plugging it back into one of the original equations finds \(a_1\) effortlessly.

Arithmetic Sequence Formula

The arithmetic sequence formula \( a_n = a_1 + (n-1)d \) defines the n-th term as a function of the first term \(a_1\), the common difference \(d\), and the term position \(n\). It is not just useful for calculation; it provides insights into the pattern of the sequence.
In our specific task, we used this formula to express the given terms \(a_5\) and \(a_{18}\) in terms of \(a_1\) and \(d\). Doing so bridges the known (term positions and their values) with the unknowns \(a_1\) and \(d\), allowing us to systematically solve for them.
Understanding this formula helps decipher any arithmetic sequence, whether it involves small, large, known, or unknown term numbers. It's a "guiding light" for anyone wanting to explore or solve puzzles related to sequences.