A statement Sn about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true. Sn: 1 + 3 + 5 + ... + (2n - 1) = n2

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 1

Chapter 9, Problem 1

A statement S_n about the positive integers is given. Write statements S₁, S₂ and S₃ and show that each of these statements is true. S_n: 1 + 3 + 5 + ... + (2n - 1) = n²

Verified step by step guidance

Step 1: Understand the statement S_n, which says that the sum of the first n odd numbers is equal to n squared. In other words, $1 + 3 + 5 + \ldots + (2n - 1) = n^2$.

Step 2: Write out the statements for $S_1$, $S_2$, and $S_3$ by substituting $n = 1, 2, 3$ respectively: $S_1: 1 = 1^2$ $S_2: 1 + 3 = 2^2$ $S_3: 1 + 3 + 5 = 3^2$

Step 3: Verify $S_1$ by calculating the left side and right side separately: Left side: 1 Right side: $1^2 = 1$ Since both sides are equal, $S_1$ is true.

Step 4: Verify $S_2$ by calculating the left side and right side separately: Left side: $1 + 3 = 4$ Right side: $2^2 = 4$ Since both sides are equal, $S_2$ is true.

Step 5: Verify $S_3$ by calculating the left side and right side separately: Left side: $1 + 3 + 5 = 9$ Right side: $3^2 = 9$ Since both sides are equal, $S_3$ is true.

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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 for all positive integers. It involves proving a base case (usually for n=1) and then showing that if the statement holds for an arbitrary integer k, it also holds for k+1. This method confirms the truth of infinite sequences of statements.

Arithmetic Series and Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The given series 1 + 3 + 5 + ... + (2n - 1) is an arithmetic sequence of odd numbers with a common difference of 2. Understanding how to sum such sequences is essential to analyze the problem.

Formula for the Sum of the First n Odd Numbers

The sum of the first n odd numbers is equal to n squared (n²). This formula can be derived or proven using induction or by recognizing the pattern in the sums. It is the key result that the problem asks to verify for specific values of n.

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A statement S_n about the positive integers is given. Write statements S₁, S₂ and S₃ and show that each of these statements is true. S_n: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2