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.

Verified step by step guidance

1. Identify the given statement \(S_n\): "2 is a factor of \(n^2 - n + 2\)", which means \(n^2 - n + 2\) is divisible by 2 for positive integers \(n\).
2. Write the statement \(S_k\) by substituting \(n\) with \(k\): \(S_k\): 2 is a factor of \(k^2 - k + 2\).
3. Write the statement \(S_{k+1}\) by substituting \(n\) with \(k+1\): \(S_{k+1}\): 2 is a factor of \((k+1)^2 - (k+1) + 2\).
4. Simplify the expression in \(S_{k+1}\): expand \((k+1)^2\) to get \(k^2 + 2k + 1\), then substitute and simplify:
5. \[ (k+1)^2 - (k+1) + 2 = k^2 + 2k + 1 - k - 1 + 2 \]
6. Combine like terms in the expression to get the simplified form of \(S_{k+1}\): \(k^2 + k + 2\).

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of a statement for all positive integers. It involves proving a base case (usually for n=1 or n=2) and then showing that if the statement holds for an integer k, it also holds for k+1. This method is essential for understanding how to write and simplify statements S_k and S_{k+1}.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in a simpler or more manageable form without changing their value. This includes expanding, factoring, and combining like terms. Simplifying S_{k+1} completely requires applying these techniques to the expression n^2 - n + 2 when n = k+1.

Divisibility and Factors

Divisibility refers to whether one integer is a factor of another, meaning it divides the number without leaving a remainder. Understanding how to determine if 2 is a factor of an expression like n^2 - n + 2 is crucial. This often involves evaluating the expression modulo 2 or analyzing parity (even or odd nature) of terms.