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Chapter 1: Problem 13

Find the next term in list. \(2,6,18,54,162, \dots\)

Short Answer

Expert verified
The next term in the sequence is 486.

Step by step solution

01

Identify the pattern

Observe the sequence given: 2, 6, 18, 54, 162. The ratio of each term to the previous term is consistent. Divide a term by its predecessor to verify this: \( \frac{6}{2} = 3 \), \( \frac{18}{6} = 3 \), \( \frac{54}{18} = 3 \), and \( \frac{162}{54} = 3 \). This suggests a common ratio of 3, indicating a geometric sequence.
02

Use the geometric sequence formula

In a geometric sequence, each term is the product of the previous term and a common ratio. Since the pattern is multiplying by 3, the formula for the sequence is \( a_n = a_{n-1} \times r \), where \( r \) is the common ratio (3), and \( a_{n-1} \) is the previous term.
03

Calculate the next term

The last known term in the sequence is 162. To find the next term, multiply this by the common ratio: \( 162 \times 3 = 486 \).

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Key Concepts

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

Understanding the Common Ratio
In a geometric sequence, identifying the common ratio is essential. A common ratio is a constant factor between consecutive terms of a sequence. You discover it by dividing any term by its preceding term.
For example, in the sequence provided, 2, 6, 18, 54, 162, you can verify the common ratio by performing the following calculations:
  • \(\frac{6}{2} = 3\)
  • \(\frac{18}{6} = 3\)
  • \(\frac{54}{18} = 3\)
  • \(\frac{162}{54} = 3\)
Each of these results is 3, confirming a constant increase factor of 3, which is your common ratio. Once you recognize this, you're well on your way to understanding your sequence and predicting its future terms.
Decoding Sequence Patterns
Recognizing sequence patterns in mathematical sequences is an essential skill. They allow you to predict and extend the sequences logically. In geometric sequences, the focus is typically on multiplication.
Each term is consistent with the previous by multiplying with the same number, known as the common ratio. For instance, in the given sequence, every term is 3 times larger than the term before it.
Understanding this multiplication pattern empowers you to not only complete the sequence but also grasp the underlying rule governing it. With practice, spotting these patterns becomes intuitive as you learn to look for the constant multiplier.
Calculating the Next Term
Once you have the common ratio and understand the pattern, calculating the next term becomes straightforward. In a geometric sequence, you determine the next term by multiplying the last known term by the common ratio. Let's apply this to our sequence:
The last term we know is 162, and our common ratio is 3. Therefore, the calculation for the next term is:
  • \(162 \times 3 = 486\)
Just like that, you've extended the sequence with precise logic. Understanding this method allows you to predict upcoming terms efficiently and confirms the sequence's integrity. This concept is useful in solving numerous math problems that involve pattern recognition and prediction.

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