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 24
Find 3 + 6 + 9 + ... + 300, the sum of the first 100 positive multiples of 3.
Verified step by step guidance1
Recognize that the series 3 + 6 + 9 + ... + 300 is an arithmetic sequence where each term increases by a common difference of 3.
Identify the first term \( a_1 = 3 \) and the last term \( a_n = 300 \).
Determine the number of terms \( n \). Since the problem states the first 100 positive multiples of 3, \( n = 100 \).
Use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} (a_1 + a_n) \], where \( S_n \) is the sum of the first \( n \) terms.
Substitute the known values into the formula: \[ S_{100} = \frac{100}{2} (3 + 300) \] and simplify step-by-step to find the sum.
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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 with a constant difference between consecutive terms. In this problem, the multiples of 3 form an arithmetic sequence starting at 3 with a common difference of 3.
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Number of Terms in a Sequence
Determining the number of terms involves identifying how many elements are in the sequence. Here, the problem states the first 100 positive multiples of 3, so the sequence has exactly 100 terms.
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Sum of an Arithmetic Series
The sum of an arithmetic series can be found using the formula S = n/2 * (first term + last term), where n is the number of terms. This formula allows efficient calculation of the total sum without adding each term individually.
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Related Practice
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Evaluate each factorial expression. 18!/16!
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