/*! 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 52 $$ 2,-6,18,-54, \ldots $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 52

$$ 2,-6,18,-54, \ldots $$ No sum

Short Answer

Expert verified
The next term in the sequence is 162.

Step by step solution

01

Identify the Pattern

Observe the progression of the sequence: 2, -6, 18, -54, ...Notice that each term is obtained by multiplying the previous term by a specific number. Begin by dividing the second term by the first: \(-6 \div 2 = -3\). Similarly, check the next pair: 18 \div -6 = -3. And check the next one: \(-54 \div 18 = -3\).Each term is obtained by multiplying the previous term by \(-3\). This indicates that the sequence is a geometric sequence with a common ratio \(r = -3\).
02

Use the Formula for the nth Term

In a geometric sequence, the nth term \(a_n\) can be found using the formula:\[ a_n = a_1 \, r^{(n-1)} \]where \(a_1\) is the first term, and \(r\) is the common ratio. In this sequence, \(a_1 = 2\) and \(r = -3\).
03

Find the Next Term in the Sequence

Apply the common ratio multiplication to find the next term.The fourth term is \(-54\). To find the fifth term, multiply this term by \(-3\):\[ -54 \times (-3) = 162 \].So the next term in the sequence is 162.

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.

Common Ratio
In a geometric sequence, the common ratio is a key factor that helps us understand how each term relates to the previous one. The common ratio, often denoted as \( r \), is a constant factor that we multiply each term by to get the next term in the sequence.
In the sequence you're working with, "2, -6, 18, -54, ...," each term is generated by multiplying the preceding term by \(-3\).
This constant, \(-3\), is your common ratio.
  • Find the common ratio by dividing any term by the one before it.
  • Example: \(-6 \div 2 = -3\), \( 18 \div -6 = -3\).
  • It must remain constant throughout the sequence.
Identifying the common ratio is crucial because it underpins the repeating pattern of the sequence.
Nth Term Formula
The nth term formula for a geometric sequence allows us to find any term without needing to calculate all the previous ones. The formula is:\[ a_n = a_1 \, r^{(n-1)} \]Here, \(a_n\) represents the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.
For our sequence, the first term \(a_1\) is 2, and the common ratio \(r\) is \(-3\).
Let's see how it works in practice:
  • Plug the values into the formula: \(a_n = 2 \times (-3)^{(n-1)}\).
  • To find the third term: \(a_3 = 2 \times (-3)^{2} = 2 \times 9 = 18\).
  • Notice how repeating this process helps predict any term in the sequence.
This formula is especially helpful when you need to find a term further along without calculating each preceding term one by one.
Sequence Pattern
Understanding the pattern in a sequence is essential for recognizing the underlying mathematical relationship.
Geometric sequences are unique because they have a multiplicative pattern, created by a common ratio. Here's how to grasp the sequence pattern:
  • Start by identifying if it's a geometric sequence using the common ratio method.
  • Notice the repetition: multiply by \(r\) to advance from one term to the next.
  • A geometric sequence either grows or shrinks exponentially due to this fixed ratio.
In the given sequence, "2, -6, 18, -54, ...," you recognize the consistent multiplicative pattern by applying \(-3\) to each term.
This discovery simplifies predicting subsequent terms and understanding the sequence's behavior.

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

Math University had an enrollment of 12,800 students in 1998. Each year the enrollment decreased by 75 students. What was the enrollment in 2005 ? 12,275

A pump is attached to a container for the purpose of creating a vacuum. For cach stroke of the pump, one-third of the air remaining in the container is removed. To the nearest tenth of a percent, how much of the air remains in the container after seven strokes? \(\quad 5.9 \%\)

\frac{1}{2}, \frac{3}{8}, \frac{9}{32}, \frac{27}{128}, \ldots .2

\(9^{n}-1\) is divisible by 8 for all positive integer values for \(n\).

\text { The } 9 \text { th term of } 8,4,2,1, \ldots, \frac{1}{32}
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.