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. Sn: 3 is a factor of n^3 - n.

Key Concepts

Factorization

Factorization is the process of breaking down an expression into its constituent factors. In the context of the statement S_n, we need to express n^3 - n in a factored form to analyze its divisibility by 3. Recognizing that n^3 - n can be factored as n(n^2 - 1) = n(n - 1)(n + 1) helps in understanding how the product of these terms relates to the number 3.

Divisibility

Divisibility refers to the ability of one integer to be divided by another without leaving a remainder. In this case, we need to show that 3 divides n^3 - n for all positive integers n. By examining the factored form n(n - 1)(n + 1), we can determine that among any three consecutive integers (n, n - 1, n + 1), at least one must be divisible by 3, thus proving the statement.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves two steps: proving the base case (usually for n=1) and then showing that if the statement holds for an arbitrary integer k, it also holds for k+1. This method is particularly useful for proving statements like S_n, as it allows us to confirm the validity of the statement for all positive integers.