Problem 4 - A Graphical Approach to College Algebra

Question

Find the common difference d for each arithmetic sequence. Do not use a calculator. $$-8,-12,-16,-20, \dots$$

Step-by-Step Solution

Answer

The common difference \(d\) is \(-4\).

Step 1: Understand the Problem

First, identify that the problem asks us to find the common difference \(d\) in an arithmetic sequence.

Step 2: Recognize the Pattern

In an arithmetic sequence, the difference between consecutive terms is always constant. This constant is called the common difference \(d\). In this sequence: \(-8, -12, -16, -20, \dots \).

Step 3: Select Consecutive Terms

Choose any two consecutive terms from the sequence. Here, take \(-12\) and \(-8\) as the two consecutive terms.

Step 4: Calculate the Difference

Subtract the earlier term from the later term: \(-12 - (-8)\). This simplifies to \(-12 + 8 = -4\).

Step 5: Verify With the Next Terms

To confirm, repeat the calculation with the next pair of consecutive terms: \(-16 - (-12) = -16 + 12 = -4\). Once more, using \(-20\) and \(-16\): \(-20 - (-16) = -20 + 16 = -4\).

Step 6: Conclude the Common Difference

Since the differences are consistent, the common difference in the sequence is \(-4\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a fundamental idea that helps predict the terms in the sequence. Simply put, it's the constant difference between any two consecutive terms. Imagine you're climbing stairs that are perfectly spaced; each step is like a jump between terms.
To find this common difference, simply subtract the first term you've chosen from the second term.
Using the example sequence \[ -8, -12, -16, -20, \dots \], we can calculate the common difference by choosing two numbers. If we take -12 and -8, we can do:

• Calculate: \(-12 - (-8)\)
• Which equals \(-12 + 8 = -4\)

You'll notice that no matter which consecutive terms you pick, the difference will remain the same, proving the sequence’s common difference.

Consecutive Terms

The key to understanding an arithmetic sequence is to grasp the idea of consecutive terms. These are terms that follow one after the other without any breaks, like beads on a string.
In the sequence \(-8, -12, -16, -20, \dots\), each number directly follows the one before it. These consecutive terms make it straightforward to find patterns such as the common difference.
To identify consecutive terms, you simply pick any pair one after the last. For example, -12 and -8 are consecutive terms.
The same logic applies when you analyze -16 and -12, or -20 and -16. When dealing with consecutive terms, always ensure they're right next to each other. This will help in keeping calculations accurate and straightforward.

Arithmetic Progression

An arithmetic sequence can also be referred to as an arithmetic progression. An arithmetic progression is a sequence of numbers in which the difference of any two successive numbers is a constant known as the common difference.
This type of progression is linear, meaning it moves at a constant rate, much like a treadmill moving at a steady pace. Each new term in an arithmetic progression is formed by adding the common difference to the previous term.
For our sequence: \[-8, -12, -16, -20, \dots\] we noticed that -4 was the common difference. You go from -8 to -12 by subtracting 4, then from -12 to -16 the same way, and so on.
Understanding the pattern in arithmetic progression makes predicting future terms easy. Once you know the starting point and the common difference, you can quickly develop the sequence's terms.