In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n² - n.

Verified step by step guidance

1. Start by understanding the statement to prove: For every positive integer $n$, 2 is a factor of $n^2-n$. This means $n^2\; -\; n$ is even for all positive integers $n$.
2. Base Case: Verify the statement for $n\; =\; 1$. Substitute $n\; =\; 1$ into the expression $n^2\; -\; n$ and check if the result is divisible by 2.
3. Inductive Hypothesis: Assume the statement is true for some positive integer $k$, i.e., assume $2$ divides $k^2\; -\; k$. This means there exists an integer $m$ such that $k^2\; -\; k\; =\; 2m$.
4. Inductive Step: Prove the statement for $k\; +\; 1$. Consider the expression $(k+1)^2\; -\; (k+1)$. Expand and simplify this expression, then use the inductive hypothesis to show that it is also divisible by 2.
5. Conclude that since the base case holds and the inductive step is true, by mathematical induction, 2 divides $n^2\; -\; n$ for every positive integer $n$.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.

Divisibility and Factors

Divisibility means one integer is a factor of another if it divides it without leaving a remainder. In this problem, showing that 2 is a factor of n² - n means proving n² - n is always even for all positive integers n. Understanding how to manipulate expressions to reveal factors is key.

Algebraic Manipulation of Expressions

Algebraic manipulation involves rewriting expressions to reveal properties like factors or divisibility. For n² - n, factoring it as n(n - 1) shows the product of two consecutive integers, which is always even. Recognizing such patterns simplifies proving divisibility statements.