Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) for each arithmetic sequence. $$a_{6}=-8, a_{7}=-18$$

Step-by-Step Solution

Answer

The first term, \(a_1\), is 42.

Step 1: Introduction

We need to find the first term \(a_1\) of an arithmetic sequence using the given terms \(a_6 = -8\) and \(a_7 = -18\). An arithmetic sequence has the form \(a_n = a_1 + (n-1)d\), where \(d\) is the common difference.

Step 1: Determine the Common Difference

Calculate the common difference \(d\) using the given terms. The formula is \(d = a_{n+1} - a_n\). Substitute \(n = 6\): \(d = a_7 - a_6 = -18 - (-8) = -18 + 8 = -10\).

Step 2: Use the Sixth Term to Find the First Term

We know \(a_6 = a_1 + 5d\). Using the known value \(a_6 = -8\) and \(d = -10\), we get the equation \(-8 = a_1 + 5(-10)\). Simplifying, \(-8 = a_1 - 50\).

Step 3: Solve for the First Term

Isolate \(a_1\) in the equation \(-8 = a_1 - 50\). Add 50 to both sides to get \(a_1 = 42\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key component. It represents the constant amount you add or subtract to move from one term in the sequence to the next. This difference, denoted by \(d\), is what makes the sequence 'arithmetic'. It is consistent throughout the sequence, allowing you to predict future terms once the difference is known.
To find the common difference, simply take the difference between any two consecutive terms. In the exercise, we used \(a_6\) and \(a_7\) to determine it:

• \(a_7 = -18\) and \(a_6 = -8\)
• The common difference \(d = a_7 - a_6 = -18 - (-8)\)
• Simplifying gives \(d = -18 + 8 = -10\)

Once you know \(d\), you can use it to find other terms of the sequence and establish its pattern.

First Term Calculation

Finding the first term of an arithmetic sequence is crucial if you want to fully understand its beginning and build the entire sequence. The first term is often the starting point denoted as \(a_1\).
To calculate \(a_1\), use any known term and work backward using the common difference. In our solution, we utilized the sixth term, \(a_6 = -8\), and the formula:

• \(a_6 = a_1 + 5d\)
• Plug in the values: \(-8 = a_1 + 5(-10)\)
• Simplify and solve for \(a_1\): \(-8 = a_1 - 50\)
• Add 50 to both sides to isolate \(a_1\): \(a_1 = 42\)

Understanding how to find the first term gives you the power to reconstruct the sequence from the very beginning.

Sequence Formula

The sequence formula for an arithmetic sequence is one of the fundamental equations in understanding how these sequences work. It is written as:

• \(a_n = a_1 + (n-1)d\)

This formula helps to find any term in the sequence once the first term and the common difference are known. Here:

• \(a_n\) is any term in the sequence, the \(n\)-th term
• \(a_1\) is the first term
• \(d\) is the common difference

By using the sequence formula, you can also verify your calculations for other terms. For instance, if you know \(a_1\) and \(d\), you can predict what \(a_5\) or \(a_{10}\) might be without needing the intervening terms. This formula is powerful, as it provides a direct way to investigate and explore arithmetic sequences once parameters are established.