Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 5

Chapter 1, Problem 5

Identify the set { 1,1/3, 1/9 ,1/27, ....} as finite or infinite.

Verified step by step guidance

Observe the given set: $\{ 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots \}$. Notice the pattern in the terms.

Recognize that each term after the first is obtained by multiplying the previous term by $\frac{1}{3}$. This means the set is a geometric sequence with the first term $a = 1$ and common ratio $r = \frac{1}{3}$.

Understand that the ellipsis ($\ldots$) at the end of the set indicates the sequence continues indefinitely, without stopping at a last term.

Recall the definitions: a finite set has a limited number of elements, while an infinite set continues without end.

Conclude that since the sequence continues indefinitely, the set is infinite.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequences

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term, and sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely). Understanding the nature of the sequence helps determine its classification.

Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant ratio. For example, in the sequence {1, 1/3, 1/9, 1/27, ...}, each term is multiplied by 1/3. Recognizing this pattern is key to analyzing the sequence.

Finite vs. Infinite Sets

A finite set contains a limited number of elements, while an infinite set has no end and continues indefinitely. Determining whether a sequence is finite or infinite depends on whether the pattern stops or goes on forever, which is essential for classifying the given set.

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Related Practice

Textbook Question

Determine whether each statement is true or false. If false, correct the right side of the equation. (2/3)^-2 = (3/2)²

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Textbook Question

Perform the indicated operation, and write each answer in lowest terms. 2x/5 ∙ 10/x²

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Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. 3x^-1/3

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. In the expression 6³, 6 is the ____ , and 3 is the ____ .

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Textbook Question

Work each problem. Match each polynomial in Column I with its factored form in Column II.

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Textbook Question

Which choice shows the correct way to write $16 over 24$ in lowest terms?

$A . \frac{16}{24} = \frac{8 + 8}{8 + 16} = \frac{8}{16} = \frac{1}{2}$

$B . \frac{16}{24} = \frac{4 \cdot 4}{4 \cdot 6} = \frac{4}{6}$

$C . \frac{16}{24} = \frac{8 \cdot 2}{8 \cdot 3} = \frac{2}{3}$

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