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Chapter 19: Problem 60

Find the indicated quantities. If \(a_{1}, a_{2}, a_{3}, \ldots\) is an arithmetic sequence, explain why \(2^{a_{1}}, 2^{a_{2}}, 2^{a_{3}}, \ldots\) is a geometric sequence.

Short Answer

Expert verified
The sequence \(2^{a_1}, 2^{a_2}, 2^{a_3}, \ldots\) is geometric because the ratio between consecutive terms is constant, \(2^d\).

Step by step solution

01

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted as \(d\). Thus, if we start with \(a_1\), the sequence looks like: \(a_1, a_1 + d, a_1 + 2d, \ldots \).
02

Determining Terms of the New Sequence

Given the arithmetic sequence \(a_1, a_2, a_3, \ldots \), where \(a_n = a_1 + (n-1)d\), the terms of the sequence \(2^{a_1}, 2^{a_2}, 2^{a_3}, \ldots\) are expressed as \(2^{a_1}, 2^{a_1+d}, 2^{a_1+2d}, \ldots \).
03

Recognizing Geometric Sequence form

A geometric sequence is a sequence where the ratio of consecutive terms is constant. To determine if \(2^{a_n}\) forms a geometric sequence, calculate the ratio \(\frac{2^{a_{n+1}}}{2^{a_n}}\).
04

Calculating the Ratio Between Terms

The ratio \(\frac{2^{a_{n+1}}}{2^{a_n}}\) can be calculated as follows:\[\frac{2^{a_{n+1}}}{2^{a_n}} = \frac{2^{a_1+nd}}{2^{a_1+(n-1)d}} = \frac{2^{a_1 + nd}}{2^{a_1 + nd - d}} = 2^d\]This shows that the ratio between any two consecutive terms is a constant \(2^d\).
05

Concluding Geometric Sequence Property

Since the ratio of consecutive terms \(\frac{2^{a_{n+1}}}{2^{a_n}}\) is constant and equal to \(2^d\), the sequence \(2^{a_1}, 2^{a_2}, 2^{a_3}, \ldots\) is indeed a geometric sequence with a common ratio of \(2^d\).

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Key Concepts

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

Geometric Sequences
Geometric sequences are a fascinating type of mathematical series where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This number is known as the common ratio. A clear example of a geometric sequence is:
  • 2, 4, 8, 16, 32, ...
Here, each term is multiplied by 2 to get the next term, making 2 the "common ratio."
Geometric sequences are essential because they can model exponential growth and decay. The nth term of a geometric sequence can be found using the formula:
  • \( a_n = a_1 imes r^{n-1} \)
where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. This sequence is valuable in many real-world applications, such as calculating interest in finance or understanding growth in populations.
Common Difference
The common difference is a key characteristic of arithmetic sequences. It represents the consistent difference between consecutive terms in the sequence. To visualize this, consider an arithmetic sequence:
  • 5, 8, 11, 14, ...
In this sequence, each term increases by 3, making 3 the common difference.
Understanding the role of the common difference is crucial for constructing the sequence formula. The expression for the nth term in an arithmetic sequence is:
  • \( a_n = a_1 + (n-1) imes d \)
where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the position of the term in the sequence.
This formula helps quickly identify any term in the sequence without listing all previous terms.
Common Ratio
The common ratio is an important concept in geometric sequences, reflecting the factor by which the sequence terms multiply to progress. This factor remains consistent throughout the series.
  • For instance, in a sequence like 3, 6, 12, 24, ...
Every term is derived by multiplying the previous term by 2, so the common ratio is 2.
The common ratio can be found by dividing any term by its predecessor:
  • \( r = \frac{a_{n+1}}{a_n} \)
This makes it straightforward to identify whether a sequence is geometric.
A constant common ratio is what fundamentally defines a geometric sequence, ensuring that each term is a systematic scale-up (or scale-down) of the previous term. This orderly growth can represent real-world scenarios like predicting the expansion of investments or modeling population growth.

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Most popular questions from this chapter

Solve the given problems by use of the sum of an infinite geometric series. Explain why there is no infinite geometric series with \(a_{1}=5\) and \(S=2\).

Find the indicated quantities. A professional baseball player is offered a contract for an annual salary of \(\$ 5,000,000\) for six years. Also offered is a bonus (based on peformance) of either \(\$ 400,000\) each year, or a \(5.00 \%\) increase in salary each year. Which bonus option pays more over the term of the contract, and how much more?

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Solve the given problems by use of the sum of an infinite geometric series. The amounts of plutonium- 237 that decay each day because of radioactivity form a geometric sequence. Given that the amounts that decay during each of the first 4 days are \(5.882 \mathrm{g}, 5.782 \mathrm{g}\), \(5.684 \mathrm{g},\) and \(5.587 \mathrm{g},\) respectively, what total amount will decay?
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