Textbook Question
You are dealt one card from a standard 52-card deck. Find the probability of being dealt a queen.
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 17
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. 3, 12, 48, 192, ...
Verified step by step guidance1
Identify the first term of the geometric sequence, which is \(a_1 = 3\).
Determine the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{12}{3}\).
Write the formula for the general term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
Substitute the values of \(a_1\) and \(r\) into the formula to get the explicit formula for \(a_n\).
Use the formula to find the seventh term by substituting \(n = 7\) into \(a_n = a_1 \times r^{n-1}\).
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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, in the sequence 3, 12, 48, 192, ..., each term is multiplied by 4 to get the next term.
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Geometric Sequences - Recursive Formula
General Term Formula of a Geometric Sequence
The general term (nth term) of a geometric sequence is given by aₙ = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. This formula allows you to find any term in the sequence without listing all previous terms.
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Evaluating the nth Term
Once the general term formula is established, you substitute the desired term number (n) into the formula to calculate that specific term. For example, to find the 7th term, plug n = 7 into the formula and simplify to get a₇.
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Related Practice
Textbook Question
Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a50 when a1 = 7, d = 5.
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Textbook Question
Find the indicated term of the arithmetic sequence with first term, , and common difference, d. Find a12 when a1 = -8, d = -2
Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 22 + ... + 2n-1 = 2n - 1
Textbook Question
The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a1=4 and an=2an-1 + 3 for n≥2
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Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)4