In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_k+1 simplifying statement S_k+1 completely. Sn: 2 is a factor of n² - n + 2.

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1. Identify the given statement S_n: "2 is a factor of n^2 - n + 2" means that the expression n^2 - n + 2 is divisible by 2 for a positive integer n.
2. Write the statement S_k by substituting n with k: S_k states that 2 divides k^2 - k + 2, or mathematically, 2 | (k^2 - k + 2).
3. Write the statement S_(k+1) by substituting n with k+1: S_(k+1) states that 2 divides (k+1)^2 - (k+1) + 2, or 2 | ((k+1)^2 - (k+1) + 2).
4. Simplify the expression in S_(k+1): Expand (k+1)^2 to get k^2 + 2k + 1, then subtract (k+1) and add 2, resulting in k^2 + 2k + 1 - k - 1 + 2.
5. Combine like terms in S_(k+1): Simplify the expression to k^2 + k + 2, so S_(k+1) states that 2 divides k^2 + k + 2.

Key Concepts

Mathematical Induction

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

Divisibility and Factors

Divisibility refers to one integer being divisible by another if it can be expressed as a product without remainder. In this problem, understanding what it means for 2 to be a factor of an expression (n² - n + 2) is crucial. This means the expression must be even for all positive integers n, which involves analyzing parity and modular arithmetic.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in simpler or more revealing forms to facilitate analysis. Simplifying S_(k+1) means substituting n = k+1 into the expression and reducing it to a form that clearly shows divisibility by 2. This step is key to connecting the inductive hypothesis with the next case.