Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 88

Chapter 9, Problem 88

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

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Identify the terms of the sequence given: 1, -2, 4, -8, 16, ... and label them as

a_1 = 1

a_2 = -2

a_3 = 4

a_4 = -8

, and

a_5 = 16

Calculate the ratio

\(\frac{a_2}{a_3}\)

by dividing the second term by the third term:

\(\frac{-2}{4}\)

Calculate the ratio

\(\frac{a_1}{a_2}\)

by dividing the first term by the second term:

\(\frac{1}{-2}\)

Calculate the ratio

\(\frac{a_4}{a_3}\)

by dividing the fourth term by the third term:

\(\frac{-8}{4}\)

Calculate the ratio

\(\frac{a_5}{a_4}\)

by dividing the fifth term by the fourth term:

\(\frac{16}{-8}\)

. Then, observe the pattern in these ratios to understand the behavior of the sequence.

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Key Concepts

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

Sequences and Terms

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term, denoted as a₁, a₂, a₃, etc. Understanding how to identify and refer to terms is essential for analyzing relationships between them.

Ratio of Terms in a Sequence

The ratio between terms, such as a₂/a₃, compares the values of two terms in the sequence. Calculating these ratios helps identify patterns, especially in geometric sequences where the ratio between consecutive terms is constant.

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a fixed number called the common ratio. Recognizing this pattern allows for predicting terms and understanding the behavior of the sequence.

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Textbook Question

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is a_n = (3)5ⁿ Find a₂/a₃, a₁/a₂, a₄/a₃ and a₅/a₄ What do you observe?