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Chapter 9: Problem 2

Show that the given sequence is geometric, and find the common ratio. $$\frac{1}{7}, \frac{3}{7}, \frac{9}{7}, \ldots, \frac{1}{7}(3)^{a-1}, \ldots$$

Short Answer

Expert verified
The sequence is geometric with a common ratio of 3.

Step by step solution

01

Define a Geometric Sequence

A sequence is geometric if the ratio between consecutive terms is constant. This constant ratio is referred to as the common ratio, typically denoted by \( r \). We will determine if this sequence has such a common ratio.
02

Identify Consecutive Terms

In the given sequence \( \frac{1}{7}, \frac{3}{7}, \frac{9}{7}, \ldots \), identify consecutive terms. Consider the first term \( a_1 = \frac{1}{7} \) and the second term \( a_2 = \frac{3}{7} \).
03

Calculate the Common Ratio

The common ratio \( r \) is calculated by dividing the second term by the first term, which is \( r = \frac{a_2}{a_1} = \frac{\frac{3}{7}}{\frac{1}{7}} \).
04

Simplify the Ratio

Simplify the expression \( \frac{\frac{3}{7}}{\frac{1}{7}} \). Using the property of division of fractions, this simplifies to \( \frac{3}{7} \times \frac{7}{1} = 3 \). So, the common ratio \( r = 3 \).
05

Generalize the Sequence

The general term of a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \). Substitute \( a_1 = \frac{1}{7} \) and \( r = 3 \) into the formula. Thus, the general form of the sequence is \( a_n = \frac{1}{7} \cdot 3^{n-1} \).
06

Verify the Given Term

Verify that the expression \( \frac{1}{7}(3)^{a-1} \) fits the form of the \( n \)-th term derived. Since it matches \( \frac{1}{7} \cdot 3^{n-1} \), it confirms the sequence is geometric with common ratio 3.

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Key Concepts

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

Common Ratio
In a geometric sequence, the arrangement of numbers is such that each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. This constant multiplier is a crucial element that defines the sequence's pattern. Let's use the given sequence as an example:
  • The first term, \(a_1 = \frac{1}{7}\).
  • The second term, \(a_2 = \frac{3}{7}\).
To determine the common ratio \(r\), you simply divide the second term by the first term:\[r = \frac{a_2}{a_1} = \frac{\frac{3}{7}}{\frac{1}{7}}.\]Here, the common ratio \(r\) will help you understand how the sequence progresses and remains consistent.
Division of Fractions
The division of fractions can be a tricky concept, but it is made easier by a simple rule: multiplying by the reciprocal. When dividing fractions, such as \( \frac{\frac{3}{7}}{\frac{1}{7}} \), apply the reciprocal of the denominator.The reciprocal of \(\frac{1}{7}\) is \(\frac{7}{1}\). So, the division of these fractions becomes: \[\frac{3}{7} \times \frac{7}{1} = 3.\]This operation shows that dividing one fraction by another is essentially multiplying by its reciprocal.This approach ensures that fractions are handled smoothly, transforming division into an easier-to-manage multiplication task.
Sequence Term Formula
A geometric sequence can be expressed in a simple algebraic way using a sequence term formula. The formula for the \(n\)-th term of a geometric sequence is:\[a_n = a_1 \cdot r^{n-1},\]where:
  • \(a_n\) is the \(n\)-th term.
  • \(a_1\) is the first term of the sequence (\(\frac{1}{7}\) in our case).
  • \(r\) is the common ratio (which we've determined to be 3).
  • \(n\) is the term number.
Using this formula, you can generate any term of the sequence by adjusting the value of \(n\). For example, substituting \(n = 1\) gives us back the first term, and every increase in \(n\) follows the geometric progression.
Simplifying Expressions
Simplifying expressions is about reducing complex algebraic expressions into their simplest form. It involves consolidating terms, reducing fractions, and extracting common factors, making them easier to understand.In the context of our sequence, especially when finding the common ratio, simplifying \(\frac{\frac{3}{7}}{\frac{1}{7}}\) led to a straightforward result of 3.By simplifying, we:
  • Cancel out the \(7\) from both the numerator and the denominator.
  • Arrive at a cleaner expression \(\frac{3}{1} = 3\).
Understanding how to simplify complex expressions allows us to focus on the key operations leading to the final outcome. This skill is valuable not only in sequences but across many areas of mathematics.

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Most popular questions from this chapter

A company is to distribute \(\$ 46,000\) in bonuses to its top ten salespeople. The tenth salesperson on the list will receive \(\$ 1000,\) and the difference in bonus money between successively ranked salespeople is to be constant. Find the bonus for each salesperson.

The sequence defined recursively by \(x_{k+1}=x_{k} /\left(1+x_{k}\right)\) occurs in genetics in the study of the elimination of a deficient gene from a population. Show that the sequence whose \(n\) th term is \(1 / x_{n}\) is arithmetic.

Insect population The sequence defined by \(a_{k+1}=c a_{k}\left(1-a_{k}\right)\) is used in the study of insect population growth. The constant \(c\) is called the Malthusian factor. Suppose that \(1000 a_{k}\) equals the number of insects after \(k\) time intervals. If initially \(a_{1}=0.25,\) describe the behavior of the insect population for each value of \(c\) (a) \(c=0.5\) (b) \(c=1.5\) (c) \(c=2.75\)

For the given \(n\) th term \(a_{n}=f(n)\) of a sequence, use the graph of \(y=f(x)\) on the interval \([1,100]\) to verify that as \(n\) increases without bound, \(a_{n}\) approaches some real number \(c\) $$a_{n}=\left(1+\frac{1}{n}+\frac{1}{2 n^{2}}\right)^{n}$$

Chlorine levels Chlorine is often added to swimming pools to control microorganisms. If the level of chlorine rises above 3 ppm (parts per million), swimmers will experience burning eyes and skin discomfort. If the level drops below 1 ppm, there is a possibility that the water will turn green because of a large algae count. Chlorine must be added to pool water at regular intervals. If no chlorine is added to a pool during a 24 -hour period, approximately \(20 \%\) of the chlorine will dissipate into the atmosphere and \(80 \%\) will remain in the water. (a) Determine a recursive sequence \(a_{n}\) that expresses the amount of chlorine present after \(n\) days if the pool has \(a_{0}\) ppm of chlorine initially and no chlorine is added. (b) If a pool has 7 ppm of chlorine initially, construct a table to determine the first day on which the chlorine level will drop below 3 ppm.
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