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: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 2

9장, 문제 2

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

검증된 단계별 안내

Step 1: Understand the statement S_n. It says that the sum of the integers starting from 3 up to (n + 2) is equal to \( \frac{n(n + 5)}{2} \). In other words, \( 3 + 4 + 5 + \cdots + (n + 2) = \frac{n(n + 5)}{2} \).

Step 2: Write out the statements S_1, S_2, and S_3 by substituting n = 1, 2, and 3 respectively into the sum and the formula.

Step 3: For S_1, the sum is just the first term: 3. Check if \( 3 = \frac{1(1 + 5)}{2} \).

Step 4: For S_2, the sum is \( 3 + 4 \). Check if \( 3 + 4 = \frac{2(2 + 5)}{2} \).

Step 5: For S_3, the sum is \( 3 + 4 + 5 \). Check if \( 3 + 4 + 5 = \frac{3(3 + 5)}{2} \). In each case, verify that the left side equals the right side to show the statements are true.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mathematical Induction

Mathematical induction is a proof technique used to verify statements for all positive integers. It involves proving the base case (usually n=1), then assuming the statement is true for n=k, and finally proving it for n=k+1. This method ensures the statement holds for every integer n.

Summation of Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. Understanding how to express and sum such sequences is essential, as the problem involves summing consecutive integers starting from 3 up to n+2.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions and equations using algebraic rules. In this problem, it is necessary to rewrite the sum and the formula n(n+5)/2, and verify their equivalence by expanding, factoring, or rearranging terms.