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Chapter 1: Problem 10

Find the next three terms of each sequence. $$96,48,24,12, \ldots$$

Short Answer

Expert verified
The next three terms are 6, 3, and 1.5.

Step by step solution

01

Identify Pattern

First, observe the given sequence: 96, 48, 24, 12. Notice that each term is reducing by a consistent factor. Calculate the ratio between consecutive terms: \( \frac{48}{96} = \frac{1}{2}\), \( \frac{24}{48} = \frac{1}{2}\), \( \frac{12}{24} = \frac{1}{2} \). Thus, each term is half of the preceding term.
02

Calculate the Next Term

Since each term is half of the previous term, calculate the next term by multiplying the last term by \( \frac{1}{2} \). So, \(12 \times \frac{1}{2} = 6\). The next term is 6.
03

Calculate the Subsequent Terms

Continue applying the same pattern to find the following terms. Multiply 6 by \( \frac{1}{2} \) to get \(6 \times \frac{1}{2} = 3\). Similarly, for the next term, \(3 \times \frac{1}{2} = 1.5\).
04

List the Next Three Terms

Having calculated each term sequentially, the next three terms of the sequence are 6, 3, and 1.5.

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Key Concepts

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

Pattern Identification in Geometric Sequences
In geometric sequences, each term is consistently multiplied by a specific value called the common ratio. Detecting this pattern is key to understanding how the sequence continues. By analyzing the sequence 96, 48, 24, 12, we notice a regular decrease. This drop hints at a geometric pattern. To confirm, calculate the ratio between consecutive terms. Each result reveals that dividing the next term by the previous term always equals 0.5.
  • The sequence starts at 96 and halves to 48.
  • The next step is from 48 to 24, once again halving.
  • Finally, 24 halves to become 12.
This consistent pattern showcases the essence of a geometric sequence.
Ratio Calculation in Geometric Sequences
The ratio in a geometric sequence represents how much a term grows or shrinks compared to the previous term. It's crucial for predicting future terms. Let's look at the sequence 96, 48, 24, 12. To determine the ratio:
  • Divide 48 by 96, resulting in 0.5.
  • Divide 24 by 48, and you'll also get 0.5.
  • Lastly, 12 divided by 24 equals 0.5.
This consistency (0.5) confirms the common ratio. Such a number is typically a fraction or decimal when the sequence decreases, making each term smaller than its predecessor.
Term Prediction in Geometric Sequences
Predicting future terms in a sequence relies on understanding the pattern. Now, using the common ratio of 0.5 from the sequence 96, 48, 24, 12, we can find the following terms. Given the last term is 12, calculate the next one by multiplying 12 by 0.5, resulting in 6. Continuing this:
  • Multiply 6 by 0.5 to get 3.
  • Next, multiply 3 by 0.5, resulting in 1.5.
These calculations allow us to predict future terms systematically. Understanding this process may seem simple, yet it can unravel more complex sequences effectively.

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