/*! 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 7 Write the formula for \(a_{n},\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 7

Write the formula for \(a_{n},\) the general term of a geometric sequence.

Short Answer

Expert verified
The formula for \(a_n\) is \(a_n = a_1 \, r^{n-1}\).

Step by step solution

01

Identify the components of a geometric sequence

A geometric sequence is a sequence of numbers where the ratio between consecutive terms is constant. The common components are the first term, denoted as \(a_1\), and the common ratio, denoted by \(r\).
02

Write the general formula for a geometric sequence

The general formula for the \(n\)-th term of a geometric sequence is expressed as:\[ a_n = a_1 \, r^{n-1} \]This formula allows you to calculate any term in the sequence given the first term and the common ratio.

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.

General Term Formula
Geometric sequences are fascinating and easy to understand once you get the hang of the general term formula. This formula helps you figure out the value of any term in a geometric series, without having to list all the preceding terms. The formula is expressed as:\[ a_n = a_1 \, r^{n-1} \]To break it down:
  • \(a_n\) is the term you're trying to find.
  • \(a_1\) represents the first term in the sequence.
  • \(r\) stands for the common ratio, which we'll discuss in the next section.
Plug everything into the formula, and you'll have your desired term. Whether you need the fifth term or the 500th term, just substitute \(n\) with the position number you need to calculate.
Common Ratio
The common ratio is a key component in understanding geometric sequences. It's what the sequence revolves around, ensuring the pattern stays consistent. In simpler terms, the common ratio is the number you multiply by to get from one term to the next. To find it, take any term in the sequence and divide it by the previous one.\[ r = \frac{a_{n}}{a_{n-1}} \]The common ratio, \(r\), can be any real number:
  • If \(r > 1\), the sequence will grow progressively larger.
  • If \(0 < r < 1\), the sequence will shrink towards zero.
  • If \(r < 0\), the sequence will alternate signs, flipping positive and negative as it progresses.
Knowing the common ratio helps you predict the behavior of the sequence, essential for calculating further terms accurately.
First Term
The first term of a geometric sequence, denoted as \(a_1\), is the starting point and an essential ingredient in our general term formula. Without knowing the first term, it's impossible to find subsequent terms in the sequence using the formula.Here鈥檚 why the first term is crucial:
  • It's the foundation of your sequence 鈥� every term results from multiplying this starting number by the common ratio raised to a power.
  • Changing the first term changes the entire sequence, even if the common ratio remains constant.
Imagine you have a first term of 3 and a common ratio of 2. The sequence begins like this: 3, 6, 12, 24, and so on. But if the first term changes to 5 while keeping the common ratio, the sequence becomes 5, 10, 20, 40, etc. So, always remember to establish your \(a_1\) before diving into calculations.

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 each decimal in fraction form. Then check the answer by performing long division. $$ 0 . \overline{21} $$

Find the sum of each infinite geometric series, if possible. $$ \frac{18}{25}+\frac{6}{5}+2+\cdots $$

Find the sum of each infinite geometric series, if possible. $$ -54+18+(-6)+\cdots $$

Find each sum. See Example 9. $$ \sum_{k=2}^{5}(5 k) $$

Find the sum of each infinite geometric series, if possible. $$ 45+15+5+\cdots $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and 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.