Problem 4 - A Graphical Approach to Precalculus with Limits

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: Understanding Arithmetic Sequences

In an arithmetic sequence, each term after the first is obtained by adding a constant called the common difference. For the sequence given: -8, -12, -16, -20, ... We denote the terms by \(a_1, a_2, a_3, a_4\), etc.

Step 2: Identifying the Difference Between Terms

The common difference \(d\) can be found by subtracting any term from the term that follows it. Let's consider the first two terms: \(a_2 = -12\) and \(a_1 = -8\).Compute the difference: \(d = a_2 - a_1 = -12 - (-8)\).

Step 3: Calculating the Common Difference

Simplify the expression for the common difference:\(d = -12 + 8 = -4\).Thus, the common difference \(d = -4\).

Step 4: Verification

To verify, check that the difference between other consecutive terms is also \(-4\). For instance,:\(a_3 = -16\) and \(a_2 = -12\): \(-16 - (-12) = -16 + 12 = -4\). Similarly, \(a_4 = -20\) and \(a_3 = -16\):\(-20 - (-16) = -20 + 16 = -4\). All these check out, confirming that the common difference is \(-4\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the key factor that defines the gap between each term. This difference remains constant throughout the sequence.

To find the common difference \(d\), you simply take a term in the sequence and subtract the term immediately before it. For instance, if given the sequence -8, -12, -16, -20, ..., you can choose any two consecutive terms like -12 and -8.

• First term (\(a_1\)): -8
• Second term (\(a_2\)): -12
• Common difference (\(d\)): \(-12 - (-8) = -12 + 8 = -4\)

This commonality in difference helps us to quickly identify and continue the sequence effectively.

Terms of Sequence

The terms of a sequence are the individual numbers that make it up. They are typically denoted by \(a_1, a_2, a_3, ..., a_n\), where each subscript indicates the position of the term within the sequence.

In our example sequence: -8, -12, -16, -20, ..., each of these numbers is a term. The first term is -8 (\(a_1\)), the second is -12 (\(a_2\)), the third is -16 (\(a_3\)), and so on.

• First term: \(a_1 = -8\)
• Second term: \(a_2 = -12\)
• Third term: \(a_3 = -16\)
• Fourth term: \(a_4 = -20\)

Recognizing the terms as part of a sequence is crucial for understanding how the sequence behaves as a whole.

Consecutive Terms

Consecutive terms in an arithmetic sequence are terms that follow one after the other in their natural order. They help us analyze patterns and establish the constant nature of the common difference.

Taking our sequence -8, -12, -16, -20, ..., consecutive terms are terms like -8 and -12, or -12 and -16. The significance of examining consecutive terms lies in observing the uniform gap between them, the so-called common difference.

• From \(a_1 = -8\) to \(a_2 = -12\), the decrement is -4.
• Proceeding from \(a_2 = -12\) to \(a_3 = -16\), the decrement remains -4.
• Continuing from \(a_3 = -16\) to \(a_4 = -20\), the gap remains consistent at -4.

These consistent differences ensure that, as you move through the sequence, each step is predictable and follows the arithmetic rule strictly.