Textbook Question
Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
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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 49
Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
Verified step by step guidance1
Identify the arithmetic sequence given by the general term: \(a_i = 4i\), where \(i\) ranges from 1 to 100.
Write out the first three terms by substituting \(i = 1, 2, 3\) into the formula: \(a_1 = 4(1)\), \(a_2 = 4(2)\), and \(a_3 = 4(3)\).
Find the last term by substituting \(i = 100\) into the formula: \(a_{100} = 4(100)\).
Recall the formula for the sum of the first \(n\) terms of an arithmetic sequence: \(S_n = \frac{n}{2} (a_1 + a_n)\), where \(a_1\) is the first term and \(a_n\) is the last term.
Substitute \(n = 100\), the first term \(a_1\), and the last term \(a_{100}\) into the sum formula to express the sum \(S_{100}\).
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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 4, 8, 12, ..., each term increases by 4. Understanding this helps identify the pattern and calculate specific terms.
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General Term of an Arithmetic Sequence
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 and d is the common difference. This formula allows you to find any term in the sequence without listing all previous terms.
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Sum of the First n Terms of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence is given by S_n = n/2 (a_1 + a_n), where a_1 is the first term and a_n is the nth term. This formula efficiently calculates the total sum without adding each term individually.
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Related Practice
Textbook Question
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)
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Textbook Question
Use the Binomial Theorem to expand each expression and write the result in simplified form. (x3 +x-2)4
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Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...
Textbook Question
Express each repeating decimal as a fraction in lowest terms.
Textbook Question
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.
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