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Chapter 10: Problem 4

Write the first five terms of each geometric sequence. $$a_{1}=24, \quad r=\frac{1}{3}$$

Short Answer

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

Step by step solution

01

Calculate the second term (a_2)

Calculate the second term by multiplying the first term by the ratio. The formula for the nth term of a geometric sequence is \(a_n = a_1 * r^{(n-1)}\), so for the second term (n=2), it is \(a_2 = a_1 * r^{(2-1)} = 24 * (\frac{1}{3})^1 = 8\)
02

Calculate the third term (a_3)

Using the same formula, calculate the third term, where n=3. It is \(a_3 = a_1 * r^{(3-1)} = 24 * (\frac{1}{3})^2 = \frac{8}{3}\)
03

Calculate the fourth term (a_4)

Continue the process for the fourth term, where n=4. It is \(a_4 = a_1 * r^{(4-1)} = 24 * (\frac{1}{3})^3 = \frac{8}{9}\)
04

Calculate the fifth term (a_5)

Finally, calculate the fifth term where n=5. It is \(a_5 = a_1 * r^{(5-1)} = 24 * (\frac{1}{3})^4 = \frac{8}{27}\)

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Key Concepts

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

Geometric Sequence Formula
A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula for any term in a geometric sequence is crucial for understanding how these sequences work and for performing calculations efficiently.

The general formula for the nth term (\( a_n \) of a geometric sequence is \( a_n = a_1 \times r^{(n-1)} \), where:\
  • \( a_1 \) is the first term of the sequence,
  • \( r \) is the common ratio, and
  • \( n \) is the term number.
For instance, to find the second term, you'd set \( n = 2 \) in the formula and solve for \( a_2 \). This formula allows us to calculate any term within the sequence without having to know the preceding terms, which can be especially helpful for finding terms far along in a sequence.
Geometric Sequence Terms
The terms of a geometric sequence form the core of its definition. Each term is derived from the previous term and the constant ratio. When writing the terms of a geometric sequence, understanding the ramifications of the common ratio is key. If the ratio is greater than one, the terms will increase, and if it's between zero and one, they will decrease. However, if the ratio is negative, the terms will alternate between positive and negative.

For the given exercise, starting with a first term of \( a_1 = 24 \) and a common ratio of \( r = \frac{1}{3} \), the subsequent terms become progressively smaller, each being one-third of its predecessor. This pattern is inherent to the geometric nature of the sequence and is fully predictable once the first term and common ratio are known.
Arithmetic Operations in Sequences
Understanding arithmetic operations within the context of sequences is vital for manipulating and analyzing them. In arithmetic sequences, each term is the result of adding a constant value to the previous term. In geometric sequences, the equivalent operation is multiplication by a constant ratio.

When working with geometric sequences, one conducts multiplication to find subsequent terms. However, one might perform addition when summing the terms of a sequence, a common requirement in many mathematical applications. For instance, in our sequence with a common ratio of \( r = \frac{1}{3} \), we multiply but could also sum the terms we find for further analysis, like calculating the series' sum.

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Most popular questions from this chapter

Use mathematical induction to prove that each statement is true for every positive integer \(n\). $$\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$$

Use a right triangle to write \(\cos \left(\tan ^{-1} x\right)\) as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x . \quad\) (Section 4.7, Example 9).

In Exercises \(39-44\), you are dealt one card from a 52 -card deck. Find the probability that you are dealt a red 7 or a black 8 .

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I was able to find the sum of the first 50 terms of an arithmetic sequence even though I did not identify every term.

This will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}}\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?
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