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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 7
Chapter 9, Problem 7

A statement Sn about the positive integers is given. Write statements Sk and Sk+1 simplifying statement Sk+1 completely. Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

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Identify the given statement \(S_n\): the sum of the sequence \(3 + 7 + 11 + \ldots + (4n - 1)\) equals \(n(2n + 1)\).
Write the statement \(S_k\) by replacing \(n\) with \(k\): \(3 + 7 + 11 + \ldots + (4k - 1) = k(2k + 1)\).
Write the statement \(S_{k+1}\) by replacing \(n\) with \(k+1\): \(3 + 7 + 11 + \ldots + (4k - 1) + [4(k+1) - 1] = (k+1)(2(k+1) + 1)\).
Simplify the term \(4(k+1) - 1\) inside the sum: \(4k + 4 - 1 = 4k + 3\).
Simplify the right side expression \((k+1)(2(k+1) + 1)\) by expanding the parentheses: \((k+1)(2k + 2 + 1) = (k+1)(2k + 3)\).

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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 verify statements about positive integers. It involves two steps: proving the base case (usually for n=1) and then proving that if the statement holds for n=k, it also holds for n=k+1. This method confirms the statement for all positive integers.
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Arithmetic Sequences and Series

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of the first n terms forms an arithmetic series. Understanding how to express and simplify these sums is essential for working with series like 3 + 7 + 11 + ... + (4n - 1).
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Algebraic Simplification

Algebraic simplification involves rewriting expressions in simpler or more compact forms by combining like terms, factoring, or expanding. Simplifying S_{k+1} completely requires careful manipulation of algebraic expressions to clearly show the relationship between terms.
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