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
Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a200 when a1 = −40, d = 5.
Verified step by step guidance1
Identify the given information: the first term \(a_1 = -40\) and the common difference \(d = 5\).
Recall the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n - 1) \times d\).
Substitute the values into the formula for \(n = 200\): \(a_{200} = -40 + (200 - 1) \times 5\).
Simplify the expression inside the parentheses: calculate \(200 - 1\).
Multiply the result by the common difference \(5\) and then add \(-40\) to find \(a_{200}\).
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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. This constant is called the common difference, denoted by d. Understanding this helps identify the pattern and predict any term in the sequence.
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Arithmetic Sequences - General Formula
General Formula for the nth Term
The nth term of an arithmetic sequence can be found using the formula a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. This formula allows direct calculation of any term without listing all previous terms.
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Substitution and Evaluation
To find a specific term like a_200, substitute the given values of a_1, d, and n into the nth term formula. Then perform arithmetic operations carefully to evaluate the term accurately. This step is essential for solving the problem correctly.
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