/*! 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 38 Write a formula for the general ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 38

Write a formula for the general term of each infinite sequence. \(1,-3,9,-27, \dots\)

Short Answer

Expert verified
The general term is given by: a_n = (-3)^{n-1}

Step by step solution

01

Identify the pattern

Observe the given sequence: 1, -3, 9, -27, ... Note that each term alternates in sign and appears to be a multiple of the previous term by -3.
02

Express terms as powers

Rewrite the terms in terms of powers of -3: 1 = (-3)^0, -3 = (-3)^1, 9 = (-3)^2, -27 = (-3)^3, ...
03

Set up the general term

Based on the pattern identified, the general term of the sequence can be written as: a_n = (-3)^{n-1}Here, 'n' is the position of the term in the sequence.
04

Verify the formula

Check if the formula works for a few terms: For n=1: a_1 = (-3)^{1-1} = 1For n=2: a_2 = (-3)^{2-1} = -3For n=3: a_3 = (-3)^{3-1} = 9The formula correctly produces the terms 1, -3, and 9.

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.

Infinite Sequences
An infinite sequence is a list of numbers that continues forever. Unlike finite sequences, which have a fixed number of terms, infinite sequences go on endlessly. Here, our sequence is 1, -3, 9, -27, ... This sequence does not terminate and will keep adding more terms as you go further.

Understanding infinite sequences is crucial because they offer a way to describe patterns and behaviors in mathematics that do not have an endpoint. You might encounter such sequences in various mathematical contexts, including series, summations, and more advanced topics.
Pattern Identification
One of the first steps in tackling sequences is identifying the pattern. This involves looking closely at the terms given in the sequence.

Here, we observe the sequence: 1, -3, 9, -27, ... At first glance, you can see that the terms alternate in sign (positive, negative, positive, negative). Recognizing patterns like alternating signs or common ratios helps us find the formula that generates the terms.

In our case, each term is a multiple of the previous one by -3. This consistent factor indicates a clear, predictable pattern.
Powers of Numbers
To express the terms clearly, it can be helpful to use powers of numbers. In the sequence 1, -3, 9, -27, ..., one way to describe each term is through the power of a base number.

By rewriting the terms as powers of -3, we get:
  • 1 as \( (-3)^0 \)
  • -3 as \( (-3)^1 \)
  • 9 as \( (-3)^2 \)
  • -27 as \( (-3)^3 \)

This representation makes it easier to see that the term in position n is \(( -3 )^{n-1} \).
Alternating Signs
Alternating signs in sequences imply that the terms switch between positive and negative. This pattern is evident in sequences like 1, -3, 9, -27, ...

Detecting alternating signs is valuable when writing a formula. In our sequence, the sign changes consistently with each term. Using a base raised to a power, like \(( -3 )^{n-1} \), naturally incorporates the alternating sign because raising -3 to an odd power yields a negative term, while an even power yields a positive term.

By understanding these patterns, you can write a formula that accurately captures the essence of the sequence, ensuring that each term is correctly placed in the sequence with the proper sign.

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

Write the first four terms of the infinite sequence whose nth term is given. \(a_{n}=\frac{1}{n^{2}+n}\)

Write out the first four terms in the expansion of each binomial. $$\left(x^{2}+5\right)^{9}$$

A fabric designer must take into account the capability of textile machines to produce material with vertical repeats. A textile machine can be set up for a vertical repeat every \(\frac{27}{n}\) inches (in.), where \(n\) is a natural number. Write the first five terms of the sequence \(a_{n}=\frac{27}{n},\) which gives the possible vertical repeats for a textile machine.

Infestation. Leona planted 9 acres of soybeans, but by the end of each week, insects had destroyed one-third of the acreage that was healthy at the beginning of the week. How many acres does she have left after 6 weeks?

Find the sum of each series. $$\sum_{i=1}^{6} 2 i$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

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.