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Chapter 11: Problem 32

Identify each sequence as arithmetic, geometric, or neither. $$3,2,1,0,-1, \dots$$

Short Answer

Expert verified
Arithmetic sequence; common difference is -1.

Step by step solution

01

Understand the Sequence

To determine if the sequence is arithmetic, geometric, or neither, first examine the given sequence: 3, 2, 1, 0, -1, ...
02

Check for an Arithmetic Sequence

An arithmetic sequence has a common difference between consecutive terms. Calculate the differences: \[2 - 3 = -1\]\[1 - 2 = -1\]\[0 - 1 = -1\]\[-1 - 0 = -1\]All differences are the same, which is -1. Therefore, it is an arithmetic sequence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common difference
An arithmetic sequence is defined by a constant difference between consecutive terms, known as the 'common difference'. This means you subtract one term from the previous term and get the same result every time.

For example, in the sequence given in the exercise: 3, 2, 1, 0, -1, ..., each term compared with the one before it shows a consistent difference of \(-1\):
  • n\text{2} - 3 = -1

  • 1 - 2 = -1

  • 0 - 1 = -1

  • -1 - 0 = -1

This uniform subtraction confirms that the sequence has a common difference of \(-1\). This is an essential feature of arithmetic sequences.
sequence identification
Determining the type of sequence is an essential skill in precalculus. To identify a sequence, observe how each term relates to the previous one.
  • Arithmetic Sequences have a constant difference between successive terms. You'll calculate the difference like we did in the exercise (e.g., \(2 - 3 = -1\)).

  • Geometric Sequences have a consistent ratio between terms. You multiply or divide by the same number to get from one term to the next. Imagine the sequence 3, 6, 12, 24 where each term multiplies by 2.

  • Neither, for sequences that don't fit into either category, they don't have a consistent difference or ratio between terms. An example is the sequence 2, 4, 7, 11, where even though the difference changes, it's not a consistent value.

In the given problem, the common difference of \(-1 \) shows that it's an arithmetic sequence.
precalculus sequences
Precalculus covers various types of sequences and their properties, laying the foundation for calculus.
Understanding arithmetic and geometric sequences helps in modeling real-world situations, analyzing trends, and solving problems involving series and sums.
  • Arithmetic Sequences: Useful in situations involving regular, incremental changes, like savings plans or measuring time intervals.

  • Geometric Sequences: Applied in cases of exponential growth or decay, such as population growth or radioactive decay.

  • Complex Sequences: More intricate sequences might appear complex but breaking them down using fundamental properties helps. For instance, by identifying an underlying pattern or relation between terms, problems become easier to solve.

Mastering these concepts in precalculus is crucial as they frequently reappear in more advanced topics. Build a strong intuitional understanding by practicing different types of sequences.

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Most popular questions from this chapter

Determine whether the sequence \(8,4,2,1, \ldots\) is arithmetic, geometric, or neither.

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