Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ... + 1/(n(n+1)) = n/(n + 1)
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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 23
Evaluate each expression.
Verified step by step guidance1
Identify the given expression: \(1 - \frac{3P_2}{4P_3} \times \frac{3}{4}\), where \(nP_r\) represents a permutation.
Recall the formula for permutations: \(nP_r = \frac{n!}{(n-r)!}\).
Calculate \$3P_2$ using the formula: \(3P_2 = \frac{3!}{(3-2)!} = \frac{3!}{1!}\).
Calculate \$4P_3$ using the formula: \(4P_3 = \frac{4!}{(4-3)!} = \frac{4!}{1!}\).
Substitute the values of \$3P_2\( and \)4P_3$ back into the expression, simplify the fraction, multiply by \(\frac{3}{4}\), and then subtract the result from 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Permutations
Permutations refer to the number of ways to arrange a subset of items from a larger set, where order matters. The notation nPr represents the number of permutations of n items taken r at a time, calculated as n! / (n - r)!.
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Introduction to Permutations
Factorials
A factorial, denoted by n!, is the product of all positive integers from 1 up to n. Factorials are fundamental in calculating permutations and combinations, as they help determine the total number of arrangements or selections.
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5:22Factorials
Order of Operations
The order of operations dictates the sequence in which mathematical operations are performed: parentheses first, then exponents, followed by multiplication and division (left to right), and finally addition and subtraction (left to right). Correct application ensures accurate evaluation of expressions.
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Guided course
8:38Performing Row Operations on Matrices
Related Practice
Textbook Question
In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. an=−2(n−1)!
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Textbook Question
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. 1, 5, 9, 13,...
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Textbook Question
Evaluate each factorial expression. 17!/15!
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Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)5
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Textbook Question
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. 0.0004, - 0.0004, 0.04, - 0.04, ...
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