Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x3 − 1)4
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 22
Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...
Verified step by step guidance1
Identify the first term \( a_1 \) of the arithmetic sequence. Here, \( a_1 = 5 \).
Determine the common difference \( d \) by subtracting the first term from the second term: \( d = 12 - 5 = 7 \).
Use the formula for the \( n \)-th term of an arithmetic sequence: \( a_n = a_1 + (n - 1)d \). Substitute \( n = 22 \) to find the 22nd term \( a_{22} \).
Apply the formula for the sum of the first \( n \) terms of an arithmetic sequence: \[ S_n = \frac{n}{2} (a_1 + a_n) \]. Substitute \( n = 22 \), \( a_1 = 5 \), and the value of \( a_{22} \) found in the previous step.
Simplify the expression to find the sum of the first 22 terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Sequence
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. In this sequence, the difference between consecutive terms is constant, which helps identify the pattern and find any term.
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Common Difference
The common difference is the fixed amount added to each term to get the next term in an arithmetic sequence. It is found by subtracting any term from the following term, and it is essential for determining the nth term and the sum of terms.
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Sum of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence can be calculated using the formula S_n = n/2 * (first term + last term). This formula simplifies adding many terms by using the number of terms and the first and last terms of the sequence.
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Related Practice
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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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