Domanda del libro di testo
Evaluate each expression.
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 9, Problema 22
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)!
Guida verificata passo dopo passo1
Identify the general term of the sequence given by the formula \(a_n = -2 (n - 1)!\), where \(n\) is the term number and \((n - 1)!\) denotes the factorial of \((n - 1)\).
Recall that the factorial of a non-negative integer \(k\), written as \(k!\), is the product of all positive integers from 1 up to \(k\). For example, \(0! = 1\), \(1! = 1\), \(2! = 2 \times 1 = 2\), \(3! = 3 \times 2 \times 1 = 6\), and so on.
Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula \(a_n = -2 (n - 1)!\):
For each \(n\), compute \((n - 1)!\) and then multiply by \(-2\) to find \(a_n\):
Write down the values of \(a_1, a_2, a_3,\) and \(a_4\) as the first four terms of the sequence.
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Factorial Notation
The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, 0! = 1. Factorials grow rapidly and are commonly used in sequences and combinatorics.
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5:22
Factorials
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 without listing all previous terms. For example, a_n = -2(n-1)! means each term depends on the factorial of (n-1), scaled by -2.
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4:45Geometric Sequences - General Formula
Evaluating Sequence Terms
To find specific terms of a sequence, substitute the term number n into the general term formula. For a_n = -2(n-1)!, calculate (n-1)! first, then multiply by -2. This process helps write out the first few terms explicitly.
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8:22Introduction to Sequences
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