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 n^2 - n + 2.

Verified step by step guidance

1. Identify the given statement S_n: 2 is a factor of n^2 - n + 2.
2. Write the statement S_k: 2 is a factor of k^2 - k + 2.
3. Substitute k+1 for n in the expression to find S_(k+1): (k+1)^2 - (k+1) + 2.
4. Simplify the expression for S_(k+1): Expand (k+1)^2 to get k^2 + 2k + 1, then simplify the entire expression.
5. Combine like terms in the expression for S_(k+1) and check if 2 is a factor of the resulting expression.

Key Concepts

Factors

A factor of a number is an integer that can be multiplied by another integer to yield that number. In the context of the question, understanding factors is crucial because the statement Sn asserts that 2 is a factor of the expression n^2 - n + 2. This means that when n^2 - n + 2 is evaluated for positive integers, the result should be divisible by 2.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of statements, typically involving positive integers. It consists of two steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and proves it for n=k+1. This method is essential for analyzing statements like Sn in the question.

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically in the form ax^2 + bx + c. In the given statement Sn, the expression n^2 - n + 2 is quadratic. Understanding how to manipulate and evaluate quadratic expressions is vital for simplifying S_k and S_(k+1) and determining the validity of the factorization involving 2.