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Chapter 4: Problem 9

Find the next three terms of each sequence. \(9,6,3,0, \ldots\)

Short Answer

Expert verified
The next three terms are -3, -6, and -9.

Step by step solution

01

Identify Pattern

To find the next terms, we must identify the pattern in the sequence. Start by checking the difference between consecutive terms: \( 6 - 9 = -3 \) \( 3 - 6 = -3 \) \( 0 - 3 = -3 \). Thus, this is an arithmetic sequence where each term decreases by 3.
02

Apply the Pattern

Using the pattern identified (subtract 3 from each term to get the next one), we can find the next terms:1. From 0, subtract 3 to get the next term: \( 0 - 3 = -3 \). 2. From -3, subtract 3 to get the next term:\( -3 - 3 = -6 \).3. From -6, subtract 3 to get the next term:\( -6 - 3 = -9 \).
03

Write the Sequence

The next three terms of the sequence, after 0, are -3, -6, and -9.

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Key Concepts

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

Term Difference
In arithmetic sequences like the one given as an example, the term difference is essential in determining how the sequence progresses. The term difference is the amount by which we increase or decrease from one term to the next. In the provided exercise, the sequence is given as 9, 6, 3, 0, and so on. When we calculate each difference, we find:
  • From 9 to 6: The difference is 6 minus 9, which is -3.
  • From 6 to 3: Again, the difference is 3 minus 6, giving us -3.
  • From 3 to 0: The difference remains -3.
This consistent subtraction of 3 confirms that the sequence is arithmetic, where each term changes by a fixed amount of -3. Understanding this term difference is key in uncovering the repeated operation within the sequence.
Mathematical Patterns
Arithmetic sequences are one of the simplest mathematical patterns encountered in mathematics, characterized by a constant difference between terms. This structure makes them predictable and easy to analyze, especially helpful when studying further ideas in sequences and series. In our example, the sequence follows the pattern:
  • Start from the first term, 9.
  • Subtract 3 repeatedly to get subsequent terms.
This straightforward pattern means that to move forward in the sequence, we simply replicate the operation (subtracting the term difference) for each pair of terms. Recognizing such patterns helps in understanding not only arithmetic sequences but also offers tools for predicting future terms in any patterned collection of numbers.
Sequence Prediction
Sequence prediction involves using the established pattern of a sequence to find future terms. Once the pattern is clear, it becomes a tool for forecasting unknown parts of the sequence. In the exercise sequence, having identified the term difference as -3, we can continue the sequence after the last known term, 0, by:
  • Subtracting 3 from 0 to reach the first unknown term, -3.
  • Subtracting 3 again from -3 to get -6.
  • Repeating the process to move from -6 to -9.
This prediction process proves the power of understanding simple arithmetic operations in sequences, making complex problems manageable with basic arithmetic. Knowing how to predict terms efficiently can be very useful for numerous mathematical and real-world applications.

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