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

Verified step by step guidance

1. Step 1: **Base Case** - Verify the statement for the initial value, usually \( n = 1 \). Substitute \( n = 1 \) into the expression \( n^2 - n \) and check if 2 is a factor.
2. Step 2: **Inductive Hypothesis** - Assume the statement is true for some positive integer \( k \), i.e., assume \( 2 \mid (k^2 - k) \). This means \( k^2 - k = 2m \) for some integer \( m \).
3. Step 3: **Inductive Step** - Prove the statement for \( n = k + 1 \). Substitute \( n = k + 1 \) into the expression to get \((k+1)^2 - (k+1)\).
4. Step 4: **Simplify** - Expand and simplify the expression \((k+1)^2 - (k+1)\) to \(k^2 + 2k + 1 - k - 1\), which simplifies to \(k^2 + k\).
5. Step 5: **Use Inductive Hypothesis** - Notice that \(k^2 + k = (k^2 - k) + 2k\). Since \(k^2 - k\) is divisible by 2 by the inductive hypothesis, and \(2k\) is clearly divisible by 2, conclude that \(k^2 + k\) is divisible by 2, completing the induction.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of a statement for all positive integers. It involves two main 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 an arbitrary positive integer k and then proves it for k+1. This method effectively demonstrates that if the statement is true for one integer, it must be true for all subsequent integers.

Factors and Divisibility

A factor of a number is an integer that divides that number without leaving a remainder. In the context of the given statement, we are interested in showing that 2 is a factor of the expression n^2 - n for all positive integers n. This involves understanding how to manipulate algebraic expressions and apply the definition of divisibility to confirm that the result is indeed an even number, which is divisible by 2.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions using various mathematical operations. In this problem, one must factor the expression n^2 - n to show that it can be expressed in a form that clearly reveals its divisibility by 2. Recognizing that n^2 - n can be factored as n(n - 1) helps in understanding that the product of two consecutive integers (n and n-1) is always even, thus confirming that 2 is a factor.