In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of n(n + 1)(n + 2).

Verified step by step guidance

1. Step 1: Understand the statement to prove: For every positive integer $n$, the product $n(n+\; 1)(n+\; 2)$ is divisible by 6. This means 6 is a factor of $n(n\; +\; 1)(n\; +\; 2)$.
2. Step 2: Base Case: Verify the statement for $n\; =\; 1$. Calculate $1\; \backslash times\; 2\; \backslash times\; 3\; =\; 6$. Since 6 is divisible by 6, the base case holds.
3. Step 3: Inductive Hypothesis: Assume the statement is true for some positive integer $k$, i.e., assume $6\; \backslash mid\; k(k\; +\; 1)(k\; +\; 2)$ (6 divides the product).
4. Step 4: Inductive Step: Prove the statement for $k\; +\; 1$. Consider the product $(k\; +\; 1)(k\; +\; 2)(k\; +\; 3)$. You need to show that 6 divides this product.
5. Step 5: Use properties of divisibility and the fact that among any three consecutive integers, one is divisible by 2 and one is divisible by 3, to argue that $(k\; +\; 1)(k\; +\; 2)(k\; +\; 3)$ is divisible by 6, completing the induction.

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.

Divisibility and Factors

Divisibility means one integer is a factor of another if it divides it without leaving a remainder. Understanding factors helps in proving statements about multiples, such as showing that a product of terms is divisible by a certain number, like 6 in this problem.

Properties of Consecutive Integers

Consecutive integers are numbers that follow each other in order, differing by 1. Their product often has special divisibility properties because among any three consecutive integers, one is divisible by 2 and one by 3, which is key to proving divisibility by 6.