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 9
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 a8 when a1 = 6, r = 2
Verified step by step guidance1
Recall the formula for the nth term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Identify the given values: the first term \(a_1 = 6\), the common ratio \(r = 2\), and the term to find is the 8th term, so \(n = 8\).
Substitute the known values into the formula: \(a_8 = 6 \times 2^{8-1}\).
Simplify the exponent: \(8 - 1 = 7\), so the expression becomes \(a_8 = 6 \times 2^7\).
Evaluate \$2^7\( (which is \(2\) multiplied by itself 7 times) and then multiply the result by 6 to find \)a_8$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Sequence
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, if the first term is 6 and the ratio is 2, the sequence is 6, 12, 24, and so on.
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Geometric Sequences - Recursive Formula
General Term Formula of a Geometric Sequence
The nth term of a geometric sequence can be found using the formula a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. This formula allows direct calculation of any term without listing all previous terms.
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Exponentiation in Sequences
Exponentiation is used in the general term formula to raise the common ratio to the power of (n-1). Understanding how to compute powers is essential to correctly find terms in geometric sequences, such as calculating 2^(7) when finding the 8th term.
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Related Practice
Textbook Question
In Exercises 5–10, a statement Sn about the positive integers is given. Write statements Sk and Sk+1 simplifying statement Sk+1 completely. Sn: 2 is a factor of n2 - n + 2.
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Textbook Question
Evaluate the given binomial coefficient.
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Textbook Question
Write the first four terms of each sequence whose general term is given. an=2n/(n+4)
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Textbook Question
In Exercises 8–9, find each indicated sum. This is a summation, refer to the textbook.
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Textbook Question
Write the first five terms of each geometric sequence. an = - 5a(n-1), a1 = - 6
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