Textbook Question
In Exercises 11–16, a die is rolled. Find the probability of getting an odd number.
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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 11
Use mathematical induction to prove that each statement is true for every positive integer n. 4 + 8 + 12 + ... + 4n = 2n(n + 1)
Verified step by step guidance1
Identify the statement to prove using mathematical induction: For every positive integer \(n\), the sum \(4 + 8 + 12 + \ldots + 4n\) equals \(2n(n + 1)\).
Base Case: Verify the statement for \(n = 1\). Substitute \(n = 1\) into both sides of the equation and check if they are equal.
Inductive Hypothesis: Assume the statement is true for some positive integer \(k\), that is, assume \(4 + 8 + 12 + \ldots + 4k = 2k(k + 1)\) holds.
Inductive Step: Using the inductive hypothesis, prove the statement is true for \(k + 1\). Start with the left side for \(n = k + 1\): \(4 + 8 + 12 + \ldots + 4k + 4(k + 1)\).
Show that adding \(4(k + 1)\) to the sum \(2k(k + 1)\) (from the inductive hypothesis) simplifies to \(2(k + 1)((k + 1) + 1)\), which matches the right side of the formula for \(n = k + 1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mathematical Induction
Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.
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Arithmetic Series
An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, the series 4 + 8 + 12 + ... + 4n has a common difference of 4. Understanding how to express and sum such series is essential to verify the formula given.
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Formula for the Sum of an Arithmetic Series
The sum of the first n terms of an arithmetic series can be calculated using the formula S_n = n/2 (first term + last term). Applying this formula helps to derive or verify the closed-form expression 2n(n + 1) for the given series, which is crucial for the induction proof.
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Related Practice
Textbook Question
Use the formula for nCr to evaluate each expression. 7C7
Textbook Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a12 when a1 = 5, r = - 2
Textbook Question
In Exercises 11–16, a die is rolled. Find the probability of getting a 4.
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Textbook Question
Write the first six terms of each arithmetic sequence. an = an-1 -10, a1 = 30
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Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form.