/*! 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 56 Write each national number in th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 56

Write each national number in the form of an infinite geometric series. $$\frac{23}{99}$$

Short Answer

Expert verified
The number \(\frac{23}{99}\) is an infinite geometric series: \(0.23 + 0.23 \times 0.01 + 0.23 \times 0.01^2 + ... \).

Step by step solution

01

Recognize the Decimal Representation

The fraction \( \frac{23}{99} \) is a repeating decimal. To find the decimal form, divide 23 by 99, which gives 0.232323... with "23" repeating indefinitely.
02

Express the Repeating Decimal

The repeating decimal 0.232323... can be expressed as 0.23 + 0.0023 + 0.000023 + ... Each term in this sequence is obtained by multiplying the previous term by 0.01.
03

Identify the Common Ratio

Observe that each subsequent term in the series is 0.01 times the previous term. Thus, the common ratio \( r \) of this geometric series is 0.01.
04

Write the Infinite Geometric Series

The decimal 0.232323... can be represented as an infinite geometric series \( 0.23 + 0.23 imes 0.01 + 0.23 imes 0.01^2 + ... \). This showcases the series where each consecutive term is 0.23 multiplied by increasing powers of 0.01.

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.

Repeating Decimal
A repeating decimal is a decimal number that continues forever while repeating a pattern of digits. In the case of the fraction \( \frac{23}{99} \), when we divide 23 by 99, the result is a decimal where the digits "23" repeat indefinitely.
  • The process of converting a fraction to its decimal form involves division, which in this case results in 0.232323... .
  • This repeating sequence of '23' highlights that after a certain point, the decimal pattern begins again, infinitely.
With repeating decimals, it's important to not only recognize the pattern but also understand how these can be translated into other mathematical forms, such as a geometric series.
Common Ratio
The common ratio is a key component of a geometric series, representing how each term in the series relates to the one before it. When dealing with repeating decimals like 0.232323..., the decimal can be expressed in terms of a geometric series.
  • In the series for 0.232323..., each subsequent term is formed by multiplying the number before it by 0.01.
  • For instance, after starting with 0.23, multiplying by 0.01 gives us the next term: 0.0023.
This repetitive multiplication by 0.01 designates 0.01 as the common ratio \( r \). Understanding the common ratio is crucial as it helps us to foresee how each term will evolve, growing smaller at a predictable rate as the sequence progresses.
This consistent rate of change is what allows a repeating decimal to transform into a geometric series conceptually.
Geometric Series Representation
A geometric series is a sequence where each term after the first is found by multiplying the previous term by a constant, known as the common ratio. When representing a repeating decimal like 0.232323... as a geometric series, we want to capture the structure of its repetition.
  • The initial term of our series is 0.23.
  • Subsequent terms are obtained by multiplying this first term by the common ratio, 0.01.
  • This creates an infinite series: \( 0.23 + 0.23 \times 0.01 + 0.23 \times 0.01^2 + \ldots \).
Each term grows successively smaller because we are continuously multiplying by the common ratio, further showing how the decimal repeats. Writing 0.232323... this way gives a deeper understanding of its structure, transforming a straightforward repeating decimal into an informative mathematical series that can be analyzed and summed.

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

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. Five \(a^{\prime} s,\) three \(b^{\prime} s\)

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$2,8,32,128,512$$

Conditional Probability and Dependent Events Two dice are rolled. If the first die shows a 2, find the probability that the sum of the dice is 7 or more.

Determine if \(f\) is a geometric sequence. $$f(n)=2(n-1)^{n}$$

Use Pascal's triangle to help expand the expression. $$ (m+n)^{6} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.