In Exercises 1–4, 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.

Verified step by step guidance

1. Step 1: Understand the statement S_n: "3 is a factor of n^3 - n" means that for each positive integer n, the expression n^3 - n is divisible by 3 without remainder.
2. Step 2: Write out the first three statements explicitly: S1: 3 divides 1^3 - 1, S2: 3 divides 2^3 - 2, and S3: 3 divides 3^3 - 3.
3. Step 3: Calculate each expression without simplifying fully: For S1, compute 1^3 - 1; for S2, compute 2^3 - 2; for S3, compute 3^3 - 3.
4. Step 4: Check divisibility by 3 for each result by seeing if the expression modulo 3 equals zero, i.e., verify that (n^3 - n) mod 3 = 0 for n = 1, 2, 3.
5. Step 5: Conclude that since 3 divides each of these expressions, the statements S1, S2, and S3 are true, supporting the general statement S_n.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves proving a base case (usually for n=1) and then showing that if the statement holds for an arbitrary positive integer k, it also holds for k+1. This method is essential for proving statements about all positive integers.

Divisibility and Factors

Divisibility refers to one integer being divisible by another without leaving a remainder. Saying '3 is a factor of n^3 - n' means that when n^3 - n is divided by 3, the remainder is zero. Understanding how to test and prove divisibility is crucial for verifying such statements.

Algebraic Manipulation and Factoring

Algebraic manipulation involves rewriting expressions to reveal useful properties. Factoring n^3 - n into n(n^2 - 1) = n(n-1)(n+1) shows it as a product of three consecutive integers. Recognizing this factorization helps in proving divisibility by 3, since among any three consecutive integers, one is divisible by 3.