Textbook Question
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 49
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)
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Identify the pattern in the sum: each term has the form \( \frac{i}{i+1} \), where \( i \) starts at 1 and increases by 1 for each term.
Determine the number of terms in the sum. Since the last term is \( \frac{14}{14+1} \), the index \( i \) goes from 1 to 14.
Write the summation notation using the index \( i \), the lower limit 1, and the upper limit 14, with the general term \( \frac{i}{i+1} \).
Express the sum as: \[ \sum_{i=1}^{14} \frac{i}{i+1} \]
This notation compactly represents the entire sum \( \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \cdots + \frac{14}{15} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Summation Notation
Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). It includes an index of summation, lower and upper limits, and a general term formula. This notation simplifies writing long sums and helps in analyzing series.
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Interval Notation
Index of Summation and Limits
The index of summation (commonly i) represents the variable that changes in each term of the sum. The lower limit indicates where the summation starts, and the upper limit shows where it ends. Correctly identifying these limits is essential to accurately express the sum.
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General Term of the Sequence
The general term defines the formula for each term in the sum based on the index i. For the given sum, each term is a fraction with numerator i and denominator i+1. Recognizing this pattern allows writing the sum compactly using summation notation.
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Related Practice
Textbook Question
Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
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Textbook Question
Express each repeating decimal as a fraction in lowest terms.
Textbook Question
Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
Textbook Question
Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)
Textbook Question
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.
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