Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)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 11
In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)n+1/(2n−1)
Verified step by step guidance1
Identify the general term of the sequence given by the formula: \(a_n = \frac{(-1)^{n+1}}{2^n - 1}\).
Understand that to find the first four terms, you need to substitute \(n = 1, 2, 3,\) and \(4\) into the formula separately.
Calculate each term by plugging in the values of \(n\): for each term, compute the numerator \((-1)^{n+1}\) and the denominator \(2^n - 1\).
Write each term as a fraction with the calculated numerator and denominator for \(n=1, 2, 3,\) and \(4\).
List the four terms in order: \(a_1, a_2, a_3,\) and \(a_4\) to complete the sequence.
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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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Exponents and Powers
Exponents represent repeated multiplication, such as 2^n meaning 2 multiplied by itself n times. Evaluating powers correctly is essential when calculating terms involving expressions like 2^n in the denominator.
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Alternating Signs Using (-1)^{n+1}
The factor (-1)^{n+1} causes the terms to alternate in sign because (-1) raised to an even power is positive, and to an odd power is negative. This pattern affects the sign of each term in the sequence.
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Related Practice
Textbook Question
Use the formula for nCr to evaluate each expression. 10C6
Textbook Question
Use the formula for nCr to evaluate each expression. 11C4
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Textbook Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a12 when a1 = 5, r = - 2
Textbook Question
In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17
Textbook Question
In Exercises 11–16, a die is rolled. Find the probability of getting a 4.
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