/*! 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 3 Find the common ratio for each g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 3

Find the common ratio for each geometric sequence. $$-15,30,-60,120, \dots$$

Short Answer

Expert verified
The common ratio for the geometric sequence -15, 30, -60, 120, ... is -2.

Step by step solution

01

Identification of terms

First, Identify any two consecutive terms in the given sequence (-15, 30, -60, 120,...). Let's take -15 and 30.
02

Calculate ratio

Divide the second term by the first term. In this case, divide 30 by -15. This will give us the common ratio for this geometric sequence. In mathematical terms, the ratio would be: \( r = \frac{second term}{first term} = \frac{30}{-15} \)
03

Finding Ratio

Solve the division to get the common ratio. \( r = -2\)

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 concept of a common ratio plays a crucial role. It is the constant factor between any two consecutive terms in the sequence. This means each term after the first is obtained by multiplying the previous one by this fixed number.
To find the common ratio, select two consecutive numbers from the sequence and divide the second number by the first.
In our example, the sequence \(-15, 30, -60, 120, \dots\) has the following steps:
  • Choose two consecutive terms, like -15 and 30.
  • Divide 30 (second term) by -15 (first term).
  • The result, \(-2\), is the common ratio for this sequence.
Understanding the common ratio helps us in extrapolating the sequence further or in identifying properties of the set that follows.
Consecutive Terms
Consecutive terms in a sequence matter because they are used to determine various properties of the sequence, especially when calculating the common ratio.
In a geometric sequence, like the one given, choosing consecutive terms such as -15 and 30 ensures accuracy in finding the common ratio.
Consecutive terms are simply terms that follow one after another in the series without any terms in between.
Always remember:
  • Each term is derived from the previous term by multiplying with the common ratio.
  • Choosing different sets of consecutive terms, like 30 and -60, in a consistent sequence will yield the same common ratio.
This principle of consecutive terms makes calculations straightforward and the sequence predictable.
Mathematical Terms
Mathematical terms like 'term', 'ratio', and 'sequence' provide clarity when solving problems related to sequences.
It鈥檚 vital to understand these terms for interpreting and solving sequence-related problems:
  • A 'term' refers to an individual element or number in the sequence, such as -15 or 30 in our case.
  • A 'ratio' indicates the factor by which the sequence scales up or down, which for our sequence is \(-2\).
  • A 'sequence' is simply an ordered list of numbers that follow a particular pattern, like increasing or decreasing geometrically.
Grasping these terms enhances comprehension and facilitates the handling of complex mathematical challenges involving sequences.
Division in Sequences
Division is an essential operation in understanding sequences, especially geometrical ones.
When finding the common ratio, division helps determine the consistent multiplier between terms.
Here鈥檚 a deeper look:
  • By dividing two consecutive terms, we directly assess the regular increase or decrease from one term to the next.
  • This division simplifies recognizing the pattern that defines the specific geometric sequence.
In essence, division not only simplifies calculations of a geometric progression but also serves as a powerful tool in identifying the nature and properties of sequences.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January \(10,\) how many degree-days are included from January 1 to January 10?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Rationalize the denominator: \(\frac{6}{\sqrt{3}-\sqrt{5}}\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=0}^{6}(-1)^{i}(i+1)^{2}=\sum_{i=1}^{7}(-1) j^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.