Textbook Question
Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...
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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 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)!
Verified step by step guidance1
Understand the general term of the sequence given by \(a_n = 2(n+1)!\). Here, \((n+1)!\) means the factorial of \((n+1)\), which is the product of all positive integers from 1 up to \((n+1)\).
Recall that the factorial function is defined as \(k! = k \times (k-1) \times (k-2) \times \cdots \times 1\) for any positive integer \(k\), and \(0! = 1\) by definition.
To find the first four terms, substitute \(n = 1, 2, 3, 4\) into the formula \(a_n = 2(n+1)!\) one at a time.
Calculate each term step-by-step: for each \(n\), first compute \((n+1)!\), then multiply the result by 2 to get \(a_n\).
Write down the four terms in order: \(a_1\), \(a_2\), \(a_3\), and \(a_4\) as the first four terms of the sequence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorials
A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials grow very quickly and are commonly used in sequences and combinatorics.
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General Term of a Sequence
The general term a_n of a sequence defines the nth term as a function of n. It allows you to find any term in the sequence without listing all previous terms. For example, a_n = 2(n+1)! gives a formula to compute each term directly.
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Evaluating Sequence Terms
To find specific terms of a sequence, substitute the term number n into the general term formula and simplify. For instance, to find the first four terms of a_n = 2(n+1)!, calculate a_1, a_2, a_3, and a_4 by plugging in n = 1, 2, 3, and 4.
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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
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)!
Textbook Question
Evaluate each factorial expression. 17!/15!
Textbook Question
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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