/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Find the next three terms of eac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 10

Find the next three terms of each sequence. $$96,48,24,12, \ldots$$

Short Answer

Expert verified
The next three terms are 6, 3, and 1.5.

Step by step solution

01

Identify Pattern

First, observe the given sequence: 96, 48, 24, 12. Notice that each term is reducing by a consistent factor. Calculate the ratio between consecutive terms: \( \frac{48}{96} = \frac{1}{2}\), \( \frac{24}{48} = \frac{1}{2}\), \( \frac{12}{24} = \frac{1}{2} \). Thus, each term is half of the preceding term.
02

Calculate the Next Term

Since each term is half of the previous term, calculate the next term by multiplying the last term by \( \frac{1}{2} \). So, \(12 \times \frac{1}{2} = 6\). The next term is 6.
03

Calculate the Subsequent Terms

Continue applying the same pattern to find the following terms. Multiply 6 by \( \frac{1}{2} \) to get \(6 \times \frac{1}{2} = 3\). Similarly, for the next term, \(3 \times \frac{1}{2} = 1.5\).
04

List the Next Three Terms

Having calculated each term sequentially, the next three terms of the sequence are 6, 3, and 1.5.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Pattern Identification in Geometric Sequences
In geometric sequences, each term is consistently multiplied by a specific value called the common ratio. Detecting this pattern is key to understanding how the sequence continues. By analyzing the sequence 96, 48, 24, 12, we notice a regular decrease. This drop hints at a geometric pattern. To confirm, calculate the ratio between consecutive terms. Each result reveals that dividing the next term by the previous term always equals 0.5.
  • The sequence starts at 96 and halves to 48.
  • The next step is from 48 to 24, once again halving.
  • Finally, 24 halves to become 12.
This consistent pattern showcases the essence of a geometric sequence.
Ratio Calculation in Geometric Sequences
The ratio in a geometric sequence represents how much a term grows or shrinks compared to the previous term. It's crucial for predicting future terms. Let's look at the sequence 96, 48, 24, 12. To determine the ratio:
  • Divide 48 by 96, resulting in 0.5.
  • Divide 24 by 48, and you'll also get 0.5.
  • Lastly, 12 divided by 24 equals 0.5.
This consistency (0.5) confirms the common ratio. Such a number is typically a fraction or decimal when the sequence decreases, making each term smaller than its predecessor.
Term Prediction in Geometric Sequences
Predicting future terms in a sequence relies on understanding the pattern. Now, using the common ratio of 0.5 from the sequence 96, 48, 24, 12, we can find the following terms. Given the last term is 12, calculate the next one by multiplying 12 by 0.5, resulting in 6. Continuing this:
  • Multiply 6 by 0.5 to get 3.
  • Next, multiply 3 by 0.5, resulting in 1.5.
These calculations allow us to predict future terms systematically. Understanding this process may seem simple, yet it can unravel more complex sequences effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Critical Thinking The inverse of a conditional is formed by negating both the hypothesis and conclusion of the conditional. Conditional: If it is raining, then it is cloudy. Inverse: \(\quad\) If it is not raining, then it is not cloudy. The contrapositive of a conditional is formed by negating both the hypothesis and conclusion of the converse of the conditional. Converse: If it is cloudy, then it is raining. Contrapositive: If it is not cloudy, then it is not raining. a. Write the inverse and contrapositive of this statement. If a figure has five sides, then it is a pentagon. b. Write a conditional. Then write its converse, inverse, and contrapositive.

Short Response Draw a figure that is a counterexample for the following conjecture: All figures with four sides are squares. (Lesson 1-1)

Critical Thinking Is the following statement true or false? Explain. Two rays can have at most one point in common.

Find the next three terms of each sequence. (Lesson 1-1) $$13,9,5,1, \ldots$$

Write the converse of each statement. (Lesson 1-4) If a figure is a triangle, then it has three sides.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.