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Chapter 2: Problem 57

Make a conjecture about the next item in each sequence. $$1,3,9,27$$

Short Answer

Expert verified
The next number in the sequence is 81.

Step by step solution

01

Identify the Pattern

Observe the sequence: 1, 3, 9, 27. Note that each item in the sequence can be obtained by multiplying the previous number by 3. The sequence appears to be a geometric sequence where each term after the first is the previous term multiplied by 3.
02

Calculate the Next Term

Use the pattern identified to find the next number. Since 27 is the last number in the sequence and each number is obtained by multiplying the previous number by 3, multiply 27 by 3 to get the next number: \(27 \times 3 = 81\).

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Key Concepts

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

Conjecture
When faced with a sequence of numbers, making a conjecture means proposing an idea about the future terms of the sequence based on the observed pattern. A conjecture is essentially an educated guess that is formed after careful consideration of the existing data or numbers. In our given sequence鈥�1, 3, 9, 27鈥攐ne would observe and analyze the numbers to infer the rule that governs them. In this case, since each number triples the prior number (1 becomes 3, 3 becomes 9, and so on), the conjecture would predict the next number by suggesting that the rule continues. Therefore, multiplying 27 by 3 gives us 81, which we conjecture to be the next term in our sequence.
Pattern Recognition
Pattern recognition forms the core mechanism when dealing with sequences. It involves identifying a rule or trait that consistently applies throughout a set of numbers. In the sequence 1, 3, 9, 27, a keen observation helps us see a specific pattern: each term is tripled to become the next.
Recognizing this kind of pattern:
  • Helps to understand the structure of the sequence.
  • Simplifies the process of predicting subsequent terms.
  • Identifies whether the sequence follows common types like arithmetic or geometric.
In our example, the repetitive multiplication by 3 is a clear indicator of a geometric sequence, characterizing the steady and predictable nature of the number鈥檚 growth. This clarity is what allows us to extend the sequence by logically predicting new terms.
Sequence Reasoning
Sequence reasoning is about applying logical steps to extend a sequence beyond the initial set of numbers. Once a pattern is recognized as in our sequence, 1, 3, 9, 27, sequence reasoning allows us to use this pattern consistently to anticipate further terms.
For our specific example:
  • We noted how each number is derived from its predecessor by multiplying by 3.
  • By reasoning through this repetitive process, we multiply 27 (the last shown number) by 3.
  • This calculation yields 81, the next logical number in the sequence.
Sequence reasoning doesn鈥檛 just confirm existing patterns but also empowers us to make informed predictions, verifying the validity and accuracy of our conjectures or hypotheses for what comes next in line.

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Most popular questions from this chapter

Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning.A number is divisible by 3 if it is divisible by \(6. Given: 24 is divisible by \)6 .$ Conclusion: 24 is divisible by 3 .

Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. (lesson \(2-1\) ) $$J(-3,2), K(1,8), L(5,2)$$

State the property that justifies each statement. $$\text { If } m \angle P+m \angle Q=110 \text { and } m \angle R=110 \text { , then } m \angle P+m \angle Q=m \angle R$$

State the property that justifies each statement. If $$m \angle 4=35\( and \)m \angle 5=35,\( then \)m \angle 4=m \angle 5$$

Use the following statements to write a compound statement for each conjunction and disjunction. Then find its truth value. (lesson \(2-2\) ) p: George Washington was the first president of the United States. q: A hexagon has five sides. \(r: 60 \times 3=18\) $$q \vee r$$
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