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Chapter 14: Problem 12

Write the first five terms of each geometric sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) $$a_{1}=24, r=\frac{1}{3}$$

Short Answer

Expert verified
The first five terms of the geometric sequence are: 24, 8, \(\frac{8}{3}\), \(\frac{8}{9}\), \(\frac{8}{27}\).

Step by step solution

01

Understand the Problem

We have a geometric sequence with the first term \(a_{1}\) as 24 and the common ratio \(r\) as \(\frac{1}{3}\). We need to write the first five terms of the sequence.
02

Calculate the second term

We will calculate the second term (\(a_{2}\)) by plugging in \(n = 2\) into the formula \(a_{n}\) = \(a_{1} \cdot r^{(n-1)}\). Thus \(a_{2}\) = \(24 \cdot \left(\frac{1}{3}\right)^{(2-1)}\) which equals 8.
03

Calculate the third term

Calculate the third term (\(a_{3}\)) using the same formula but replacing \(n\) with 3. Thus \(a_{3}\) = \(24 \cdot \left(\frac{1}{3}\right)^{(3-1)}\) which equals \( \frac{8}{3}\).
04

Calculate the fourth term

Same way, calculate the fourth term (\(a_{4}\)) using the formula and substituting \(n\) with 4. Thus \(a_{4}\) = \(24 \cdot \left(\frac{1}{3}\right)^{(4-1)}\) which equals \( \frac{8}{9}\).
05

Calculate the fifth term

Lastly, calculate the fifth term (\(a_{5}\)) by substituting \(n\) with 5 in the formula. Thus \(a_{5}\) = \(24 \cdot \left(\frac{1}{3}\right)^{(5-1)}\) which equals \( \frac{8}{27}\).

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Key Concepts

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

Understanding the First Term
The first term in a geometric sequence is the starting number from which the sequence is generated. It's often symbolized as \(a_1\). It serves as the foundation of the entire sequence. In our given exercise, the first term \(a_1\) is 24.
  • Imagine the first term as your starting point or the initial step in a journey of numbers.
  • Knowing the first term is crucial because every subsequent term in the sequence is built upon it.
  • Without the first term, it would be impossible to generate the rest of the sequence accurately.
The unique and fixed value of the first term sets the pace for all other terms in the sequence.
Decoding the Common Ratio
The common ratio of a geometric sequence is the factor by which we multiply the current term to get the next term. In this exercise, the common ratio \(r\) is \(\frac{1}{3}\).
  • The common ratio plays a vital role in determining how the sequence progresses.
  • If the ratio is greater than 1, the sequence will grow larger. Conversely, if it's between 0 and 1, the sequence decreases.
  • A negative common ratio means the terms will alternate between positive and negative.
In our sequence, multiplying by \(\frac{1}{3}\) means each subsequent number is reduced by two-thirds compared to the previous one, leading us on a descending journey of fractionally smaller numbers.
Exploring the Sequence Formula
The sequence formula is a mathematical expression used to find any term in a geometric sequence. The standard formula is given by:\[a_n = a_1 \cdot r^{(n-1)}\]
  • \(a_n\) represents the \(n^{th}\) term of the sequence.
  • The formula uses both the first term and the common ratio to find any term positionally.
In our exercise, using \(a_1 = 24\) and \(r = \frac{1}{3}\), we can calculate the first five terms systematically. The sequence formula is powerful because it allows for quick calculation of terms without needing to list all preceding terms.
Comprehending a Mathematical Sequence
A mathematical sequence refers to an ordered list of numbers following a specific pattern or rule. Geometric sequences are a type of mathematical sequence where each term is obtained by multiplying the previous term by a constant called the common ratio.
  • Sequences can be finite or infinite, depending on how many terms they include.
  • They are used in various fields, like computer science, finance, and physics, to model repetitive processes.
  • Beyond their mathematical utility, sequences help in understanding patterns and structures.
In our specific sequence with a starting point of 24 and a common ratio of \(\frac{1}{3}\), we see a clear mathematical structure where each term seamlessly fits into a predefined pattern, demonstrating the elegance of mathematical sequences.

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Most popular questions from this chapter

Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the exponents on \(a\).

Use the formula for the sum of the first n terms of a geometric sequence to solve. A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. \begin{aligned}&16, \quad\quad\quad\quad\quad 0.96(16), \quad(0.96)^{2}(16), \quad(0.96)^{3}(16), \ldots\\\&\begin{array}{|l|l|l|}\hline \begin{array}{l}\text { 1st swing } \\\\\end{array} & \begin{array}{l}\text{ 2nd swing } \\\\\end{array} & \begin{array}{l}\text { 3rd swing } \\\\\end{array} & \begin{array}{l}\text { 4th swing } \\\\\end{array} \\\\\hline\end{array}\end{aligned} After 10 swings, what is the total length of the distance the pendulum has swung? Round to the nearest hundredth of an inch.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. One of the terms in my binomial expansion is \(\left(\begin{array}{l}7 \\\ 5\end{array}\right) x^{2} y^{4}\).

Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.

Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.
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