Textbook Question
Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a60 when a1 = 35, d = -3.
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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 21
In Exercises 21–22, a fair coin is tossed two times in succession. The sample space of equally likely outcomes is {HH,HT,TH,TT}. Find the probability of getting two heads.
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Identify the sample space for tossing a fair coin two times. The sample space consists of all possible outcomes: \(\{HH, HT, TH, TT\}\).
Count the total number of outcomes in the sample space. Since there are two tosses and each toss has 2 possible outcomes, the total number of outcomes is 4.
Determine the favorable outcomes for the event 'getting two heads'. This means both tosses result in heads, so the favorable outcome is \(\{HH\}\).
Count the number of favorable outcomes. Here, there is only 1 favorable outcome: \(HH\).
Calculate the probability of getting two heads using the formula: \(P(\text{two heads}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} = \frac{1}{4}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sample Space
The sample space is the set of all possible outcomes of an experiment. For tossing a fair coin twice, the sample space includes {HH, HT, TH, TT}, representing all equally likely sequences of heads (H) and tails (T). Understanding the sample space is essential for calculating probabilities.
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Probability of an Event
Probability measures the likelihood of an event occurring and is calculated as the ratio of favorable outcomes to the total number of outcomes in the sample space. For example, the probability of getting two heads is the number of outcomes with two heads divided by the total outcomes.
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Equally Likely Outcomes
When all outcomes in the sample space have the same chance of occurring, they are called equally likely. This assumption allows us to use simple ratios to find probabilities, such as dividing the count of favorable outcomes by the total number of outcomes.
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Related Practice
Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x3 − 1)4
Textbook Question
Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for an to find a7, the seventh term of the sequence. 1.5, - 3, 6, -12, ...
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Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3
Textbook Question
Evaluate each expression.
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Textbook Question
In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. an=2(n+1)!
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