Problem 8 - Contemporary Precalculus

Question

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$-6,-3.7,-1.4, .9,3.2, \ldots$$

Step-by-Step Solution

Answer

Answer: The sequence is arithmetic.

Step 1: Find the Differences

Calculate the differences by subtracting previous term from the next term: \((-3.7 - -6) = 2.3\), \((-1.4 - -3.7) = 2.3\), \((0.9 - -1.4) = 2.3\), \((3.2 - 0.9) = 2.3\) Step 2: Check for a common difference.

Step 2: Common Difference

Since all the differences between consecutive terms are equal (\(2.3\)), the sequence is arithmetic. Step 3: Determine the type of the sequence.

Step 3: Type of the Sequence

The sequence is arithmetic.

Key Concepts

Common Difference

In the context of arithmetic sequences, the common difference is a crucial term to grasp. This difference refers to the constant amount that each term in an arithmetic sequence is increased (or decreased) to produce the next term. It's like adding the same spice to each dish you make, resulting in a consistent flavor.

To find the common difference, subtract any term from the term that follows it in the sequence. For instance, with the sequence \(-6, -3.7, -1.4, 0.9, 3.2, \ldots\), you start with the difference between the second term \(-3.7\) and the first term \(-6\):

• \((-3.7 - (-6)) = 2.3\)
• When you calculate this for each pair of consecutive terms, they will all be equal to \(2.3\).

Whenever you find that the differences are uniform like this, you're looking at an arithmetic sequence. Understanding this concept not only helps in identifying arithmetic sequences but also plays a vital role in predicting future values in the sequence.

Sequence Types

Sequences in mathematics can be split into various types. The most common ones are arithmetic and geometric sequences. Here, we emphasize the arithmetic sequence, which is characterized by its common difference.

• Arithmetic Sequences: These involve a fixed number being added (or subtracted) to get from one term to the next, as discussed earlier. Every term is derived by adding the common difference to the preceding term.
• Geometric Sequences: Contrasting arithmetic sequences, these involve multiplication. A constant ratio is multiplied from one term to get the next term.
• Neither: Some sequences don't fit into arithmetic or geometric categories. They don't have fixed differences or ratios.

Correctly identifying the type of sequence helps in solving complex mathematical problems and applying mathematical concepts effectively in practical scenarios.

Precalculus

Precalculus is where you dive deep into the ideas and concepts that lay the groundwork for calculus. It serves as a bridge between algebra and calculus, preparing students for more complex topics.

Within precalculus, sequences and series, specifically arithmetic sequences, are a fundamental topic. Students learn to recognize patterns, compute with them, and apply them to solve problems.

• Understanding Arithmetic Sequences: Part of the precalculus curriculum involves learning how to determine whether a sequence is arithmetic and how to solve related problems.
• Real-world Applications: These concepts are crucial, not just in theoretical problems but also in real life, like calculating interest in finance or predicting outcomes in various statistical models.

By mastering these concepts in precalculus, students prepare themselves for the vast array of possibilities in calculus and beyond, enhancing both their mathematical reasoning and problem-solving skills.