Textbook Question
In Exercises 17–20, you are dealt one card from a standard 52-card deck. Find the probability of being dealt a picture card.
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 19
Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. -7, -3, 1, 5 ...
Verified step by step guidance1
Identify the first term of the arithmetic sequence, which is given as \(a_1 = -7\).
Calculate the common difference \(d\) by subtracting the first term from the second term: \(d = -3 - (-7) = -3 + 7\).
Write the general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) \times d\).
Substitute the values of \(a_1\) and \(d\) into the formula to get the explicit formula for \(a_n\).
Use the formula to find the 20th term by substituting \(n = 20\) into the expression for \(a_n\).
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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. For example, in the sequence -7, -3, 1, 5, the common difference is 4. Understanding this pattern is essential to formulating the general term.
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Arithmetic Sequences - General Formula
General Term Formula of an Arithmetic Sequence
The general term (nth term) of an arithmetic sequence is given by an = a1 + (n - 1)d, where a1 is the first term and d is the common difference. This formula allows direct calculation of any term without listing all previous terms.
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Evaluating the nth Term
Once the general term formula is established, substituting n with the desired term number (like 20 for a20) gives the value of that term. This step involves simple arithmetic and is crucial for finding specific terms efficiently.
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Related Practice
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