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.

Verified step by step guidance

1. <Step 1: Understand the problem.> We need to verify the statement $S_n$: '3 is a factor of $n^3 - n$' for the first three positive integers, $n = 1, 2, 3$. This means we need to check if $n^3 - n$ is divisible by 3 for these values of $n$.
2. <Step 2: Calculate $S_1$.> Substitute $n = 1$ into the expression $n^3 - n$. Calculate $1^3 - 1$ and check if the result is divisible by 3.
3. <Step 3: Calculate $S_2$.> Substitute $n = 2$ into the expression $n^3 - n$. Calculate $2^3 - 2$ and check if the result is divisible by 3.
4. <Step 4: Calculate $S_3$.> Substitute $n = 3$ into the expression $n^3 - n$. Calculate $3^3 - 3$ and check if the result is divisible by 3.
5. <Step 5: Conclude the verification.> After checking $S_1$, $S_2$, and $S_3$, conclude that the statement $S_n$ is true for $n = 1, 2, 3$ as each result is divisible by 3.

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.