Textbook Question
In Exercises 17–20, you are dealt one card from a standard 52-card deck. Find the probability of being dealt a picture card.
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 19
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 = n2/n!
Verified step by step guidance1
Understand the general term of the sequence given by \(a_n = \frac{n^2}{n!}\), where \(n!\) (n factorial) means the product of all positive integers from 1 up to \(n\).
Recall that \(n! = 1 \times 2 \times 3 \times \cdots \times n\), and by definition, \(0! = 1\).
Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula \(a_n = \frac{n^2}{n!}\):
For each \(n\), compute the numerator \(n^2\) and the denominator \(n!\) separately, then divide to find \(a_n\).
Write down the terms as \(a_1 = \frac{1^2}{1!}\), \(a_2 = \frac{2^2}{2!}\), \(a_3 = \frac{3^2}{3!}\), and \(a_4 = \frac{4^2}{4!}\), simplifying each fraction as much as possible.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and General Terms
A sequence is an ordered list of numbers defined by a general term formula a_n, which gives the nth term. Understanding how to substitute values of n into the formula allows you to find specific terms in the sequence.
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Geometric Sequences - General Formula
Factorials
The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials grow rapidly and are commonly used in sequences and combinatorics.
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Evaluating Terms Involving Factorials
When a sequence term involves factorials, you must carefully compute the factorial values and perform the arithmetic operations as indicated. For a_n = n^2 / n!, calculate n^2 and divide by n! for each term to find the sequence values.
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Related Practice
Textbook Question
You are dealt one card from a standard 52-card deck. Find the probability of being dealt a queen.
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