Question

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.

Accepted Answer

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.

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.

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Key Concepts

Mathematical Induction

Factorization of Polynomials

Divisibility and Factors