Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n2
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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 13
The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a1=7 and an=an-1 + 5 for n≥2
Verified step by step guidance1
Identify the first term of the sequence, which is given as \(a_1 = 7\).
Understand the recursive formula: for each term \(a_n\) where \(n \geq 2\), the term is defined as \(a_n = a_{n-1} + 5\). This means each term is 5 more than the previous term.
Calculate the second term \(a_2\) by substituting \(n=2\) into the recursive formula: \(a_2 = a_1 + 5\).
Calculate the third term \(a_3\) by substituting \(n=3\): \(a_3 = a_2 + 5\).
Calculate the fourth term \(a_4\) by substituting \(n=4\): \(a_4 = a_3 + 5\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Recursive Sequence Definition
A recursive sequence is defined by specifying the first term(s) and a formula that relates each term to one or more previous terms. Understanding how to use the given formula to find subsequent terms is essential for generating the sequence.
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Initial Term (Base Case)
The initial term, often denoted as a₁, serves as the starting point of the sequence. It is necessary to know this value to begin applying the recursive formula and calculate the following terms.
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Iteration of the Recursive Formula
To find terms beyond the first, repeatedly apply the recursive formula by substituting the previous term's value. This step-by-step process allows you to generate the sequence terms one by one.
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Related Practice
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Use the formula for nCr to evaluate each expression. 7C7
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Use the Binomial Theorem to expand each binomial and express the result in simplified form.
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