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Chapter 4: Problem 34

What operation would you use to find the next term in the sequence \(3,6,12,24, \ldots\) ?

Short Answer

Expert verified
Multiply each term by 2.

Step by step solution

01

Analyze the Sequence

Observe the sequence: \(3, 6, 12, 24, \ldots\). Each term seems to be growing quickly, suggesting a consistent operation is being applied repeatedly to get from one term to the next.
02

Identify the Pattern

Look at the consecutive terms: \(6\) is double of \(3\), \(12\) is double of \(6\), \(24\) is double of \(12\). It appears each term is obtained by multiplying the previous term by \(2\).
03

Deduce the Operation

Since each term is double the previous term, the operation being used is multiplication. Specifically, each term is the product of the previous term and \(2\).

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Key Concepts

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

Pattern Recognition
Recognizing a pattern in a sequence is a key step in understanding sequences. It involves observing how each term changes from one to the next. In the exercise, you start by examining the numbers given: \(3,\ 6,\ 12,\ 24,\ \ldots\).

You might notice several possibilities:
  • Are the numbers increasing by the same amount? (They're not, as they double.)
  • Is there a multiplication factor applied? (Yes, each term doubles.)
Recognizing these changes helps identify the rule applied to generate the sequence. In this case, the factor is a consistent multiplication by 2. Spotting this pattern simplifies the process as each term adheres to this rule, aiding prediction of future terms.
Multiplication
Multiplication is at the heart of geometric sequences. It involves a consistent factor that transforms one term into the next. In our sequence, each number is multiplied by 2 to get to the next.

The idea is simple:
  • Start with a term, say \(n\).
  • Multiply by \(2\) to find \(2n\).
  • This new term \(2n\) then becomes the current term for multiplication.
This repetitive, simple action鈥攎ultiplying by the same number鈥攃reates a whole cascade of numbers, each one a predictable result of the previous using multiplication. It leads to an exponential growth pattern, as seen in our sequence. Multiplication here acts like a recipe, consistently growing the sequence by a fixed factor.
Sequence Analysis
Analyzing sequences requires understanding how terms are related. In our sequence \(3, 6, 12, 24, \ldots\), the relationship between terms is driven by consistent multiplication.

When analyzing any sequence, consider:
  • The starting term (here, it is \(3\)).
  • The relationship or rule applied between terms (multiplying by \(2\)).
  • Predicting future terms based on the established rule.
To extend this sequence, apply the same multiplication rule repeatedly: start with \(24\), multiply by \(2\) to get \(48\).

The beauty of sequence analysis is in recognizing these consistent transformations, making sequences both predictable and mathematically intriguing. It equips you to handle various patterns, as many natural and practical occurrences follow such structured progressions.

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Most popular questions from this chapter

Given each set of points, determine if \(\overleftrightarrow{J K}\) and \(\overleftrightarrow{L M}\) are parallel, perpendicular, or neither. \(J(6,3), K(-7,3), L(-4,-5), M(1,-5)\)

Name the slope and \(y\)-intercept of the graph of each equation. \(y=5\)

Find the next three terms of each sequence. \(10,13,19,28, \ldots\)

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

Name the slope and \(y\)-intercept of the graph of each equation. \(3 x-2 y=18\)
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