Textbook Question
In Exercises 12–15, write the first six terms of each arithmetic sequence. a1 = 3/2, d = -1/2
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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
Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n2
Verified step by step guidance1
Step 1: Understand the statement to prove by induction: For every positive integer \(n\), the sum of the first \(n\) odd numbers is equal to \(n^2\). In other words, prove that \(1 + 3 + 5 + \dots + (2n - 1) = n^2\).
Step 2: Base Case: Verify the statement for \(n=1\). Substitute \(n=1\) into the left side and right side of the equation to check if both sides are equal.
Step 3: Inductive Hypothesis: Assume the statement is true for some positive integer \(k\), that is, assume \(1 + 3 + 5 + \dots + (2k - 1) = k^2\) holds.
Step 4: Inductive Step: Using the inductive hypothesis, prove the statement is true for \(k+1\). Start with the sum up to \(k+1\) terms: \(1 + 3 + 5 + \dots + (2k - 1) + (2(k+1) - 1)\), and show this equals \((k+1)^2\).
Step 5: Simplify the expression from Step 4 by substituting the inductive hypothesis and algebraically manipulating the terms to demonstrate the equality holds, completing the induction proof.
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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.
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Sum of an Arithmetic Sequence
The sum 1 + 3 + 5 + ... + (2n - 1) is an arithmetic sequence with a common difference of 2. Understanding how to express and manipulate sums of arithmetic sequences helps in recognizing patterns and formulating the statement to be proved.
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Algebraic Manipulation
Algebraic manipulation involves simplifying expressions and substituting terms correctly. In this proof, it is essential to manipulate the inductive hypothesis and the next term to show that the sum equals n², ensuring the logical flow of the induction step.
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Related Practice
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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 a40 when a1 = 1000, r = - 1/2
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Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form.
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