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

문제 1

9장, 문제 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

검증된 단계별 안내

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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주요 개념

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Mathematical Induction

Mathematical induction is a proof technique used to verify statements 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 n=k, it also holds for n=k+1. This method confirms the statement for all natural numbers.

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 this helps in recognizing patterns and summing the terms.

Sum of the First n Odd Numbers

The sum of the first n odd numbers is equal to n squared (n^2). This is a well-known formula that can be proven by induction or visualized geometrically. It shows a direct relationship between odd numbers and perfect squares, which is the core of the given statement.