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 is a factor of n3 - n.

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 4

Capitolo 9, Problema 4

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 is a factor of n³ - n.

Guida verificata passo dopo passo

Step 1: Understand the statement S_n: "3 is a factor of n^3 - n" means that for any positive integer n, the expression n^3 - n is divisible by 3. We want to verify this for n = 1, 2, and 3, i.e., write S_1, S_2, and S_3 and check their truth.

Step 2: Write S_1 by substituting n = 1 into the expression: calculate 1^3 - 1, which simplifies to 1 - 1 = 0. Since 0 is divisible by 3, S_1 is true.

Step 3: Write S_2 by substituting n = 2 into the expression: calculate 2^3 - 2, which simplifies to 8 - 2 = 6. Since 6 is divisible by 3, S_2 is true.

Step 4: Write S_3 by substituting n = 3 into the expression: calculate 3^3 - 3, which simplifies to 27 - 3 = 24. Since 24 is divisible by 3, S_3 is true.

Step 5: Conclude that for n = 1, 2, and 3, the expression n^3 - n is divisible by 3, confirming the statements S_1, S_2, and S_3 are true. This supports the general statement S_n.

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Mathematical Induction

Mathematical induction is a proof technique used to show that a statement holds 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 is essential for validating statements defined for all natural numbers.

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Factorization of Polynomials

Factorization involves expressing a polynomial as a product of simpler polynomials or factors. Recognizing how to factor expressions like n³ - n helps identify common factors, such as 3, which is crucial for proving divisibility properties in algebraic expressions.

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Divisibility and Factors

Divisibility means one integer divides another without leaving a remainder. Understanding factors and divisibility rules, especially for integers like 3, allows us to determine whether expressions like n³ - n are multiples of 3 for all positive integers n.

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