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
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x3 − 1)4
Verified step by step guidance1
Identify the binomial expression to be expanded: \((2x^{3} - 1)^4\).
Recall the Binomial Theorem formula: \(\displaystyle (a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k\), where \(\binom{n}{k}\) is the binomial coefficient.
Set \(a = 2x^{3}\), \(b = 1\), and \(n = 4\). Write the expansion as \(\sum_{k=0}^{4} \binom{4}{k} (2x^{3})^{4-k} (-1)^k\).
Calculate each term by evaluating the binomial coefficient \(\binom{4}{k}\), raising \$2x^{3}\( to the power \)(4-k)\(, and multiplying by \)(-1)^k\( for \)k = 0, 1, 2, 3, 4$.
Simplify each term by applying the exponent to \(2\) and \(x^{3}\) separately, then combine like terms to write the final expanded polynomial.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Theorem
The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, summing k from 0 to n. This theorem simplifies the expansion process without multiplying the binomial repeatedly.
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Binomial Coefficients
Binomial coefficients, denoted as (n choose k), represent the number of ways to choose k elements from n and are calculated using factorials: n! / (k!(n-k)!). They serve as the coefficients in the expanded form of a binomial expression and determine the weight of each term.
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Exponent Rules
Exponent rules govern how to handle powers in algebraic expressions, such as multiplying powers with the same base by adding exponents and raising a power to another power by multiplying exponents. These rules are essential when expanding terms like (2x³)^m to simplify the expression correctly.
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Related Practice
Textbook Question
Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...
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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 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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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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