Textbook Question
In Exercises 11–16, a die is rolled. Find the probability of getting a number greater than 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 15
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 = 1 000 000, r = 0.1
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 \(a_1 = 1,000,000\), the common ratio \(r = 0.1\), and the term to find is \(a_8\) (the 8th term).
Substitute the known values into the formula: \(a_8 = 1,000,000 \times (0.1)^{8-1}\).
Simplify the exponent: calculate \(8 - 1\) to get 7, so the expression becomes \(a_8 = 1,000,000 \times (0.1)^7\).
Evaluate the power \((0.1)^7\) and then multiply by 1,000,000 to find the value of \(a_8\).
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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 2 and the ratio is 3, the sequence is 2, 6, 18, 54, 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 and Powers
Exponentiation involves raising a number to a power, which means multiplying the number by itself a certain number of times. In geometric sequences, powers of the common ratio determine how terms grow or shrink, making understanding exponents essential for calculating terms.
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Related Practice
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