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Chapter 9: Problem 18

Write the first five terms of the geometric sequence. $$a_{1}=2, r=\pi$$

Short Answer

Expert verified
The first five terms of the geometric sequence are: \(2, 2\pi, 2\pi^2, 2\pi^3, 2\pi^4\).

Step by step solution

01

Compute the Second Term of the Sequence

We use the geometric sequence formula with \(n=2\). Therefore \(a_{2} = a_{1} \cdot r^{(2-1)} = 2 \cdot \(\pi\)^{1} = 2\pi\).
02

Compute the Third Term of the Sequence

By substituting \(n=3\) into the formula, we get \(a_{3} = a_{1} \cdot r^{(3-1)} = 2 \cdot \(\pi\)^{2} = 2\pi^2\).
03

Compute the Fourth Term of the Sequence

Following the same method, we get \(a_{4} = a_{1} \cdot r^{(4-1)} = 2 \cdot \(\pi\)^{3} = 2\pi^3\).
04

Compute the Fifth Term of the Sequence

Finally, for the fifth term, \(a_{5} = a_{1} \cdot r^{(5-1)} = 2 \cdot \(\pi\)^{4} = 2\pi^4\).

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Key Concepts

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

Understanding a Sequence
When we talk about a sequence in mathematics, we're referring to an ordered list of numbers. Each number in the list is called a term. A sequence can be finite or infinite, depending on whether it has a limited number of terms or continues indefinitely. In the context of a geometric sequence, each term is derived by multiplying the previous term by a fixed non-zero number called the common ratio.
  • An example of a sequence could be 2, 4, 8, 16, where each term follows a rule or pattern.
  • In a geometric sequence, this rule involves multiplying by a constant.
  • The order of the numbers is crucial because it reflects the structure of the sequence.
Understanding sequences is fundamental to analyzing patterns and solving various mathematical problems. They provide a way to model and describe repetitive processes or growth over discrete steps, making them essential in fields such as finance, computer science, and natural sciences.
Grasping the Common Ratio
In a geometric sequence, the common ratio is the factor that each term is multiplied by to yield the next term. It plays a vital role in determining how the sequence progresses. This ratio can be positive, negative, or even a fraction.
  • In our example, the common ratio is \(\pi\), which indicates that each term is multiplied by \(\pi\) to get the following term.
  • To find any term in the sequence, you multiply the previous term by this common ratio.
  • If the common ratio is greater than one, the sequence will grow, while a ratio between zero and one will cause it to shrink.
The common ratio is a unique identifier of geometric sequences and is what sets them apart from arithmetic sequences, where each term is obtained by adding a fixed number. By mastering the concept of the common ratio, students can better understand and predict the behavior of sequences and variations.
Exploring Sequence Terms
Each term in a geometric sequence can be calculated using the initial term and the common ratio. The formula for finding any term \(a_n\) in a geometric sequence is:\[ a_n = a_1 \cdot r^{(n-1)} \]Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence.
  • The first term, \(a_1\), is given directly as part of the sequence.
  • To find the second term, you multiply \(a_1\) by \(r\); for the third term, multiply the previous term again by \(r\), and so on.
  • An understanding of this process allows for the calculation of any term in the sequence without having to know the previous terms.
In our exercise example, where the first term is 2 and the common ratio is \(\pi\), we applied this rule to determine the sequence's first five terms. By recognizing the relationship between terms, learners can not only solve sequencing problems but also appreciate the structure and beauty of mathematical patterns.

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