Problem 28 - Algebra and Trigonometry

Question

Use mathematical induction to prove that each statement is true for every positive integer n. 3 is a factor of \(n(n+1)(n-1)\)

Step-by-Step Solution

Answer

Yes, it's proven that 3 is a factor of \(n(n+1)(n-1)\) for every positive integer \(n\) by mathematical induction.

Step 1: Base Case

First, test the base case. For \(n=1\), check if 3 is a factor of \(1(1+1)(1-1)\). The expression simplifies to 0, and any number is a factor of 0, so 3 is indeed a factor of this expression for \(n=1\). Thus, the basis step holds.

Step 2: Inductive Hypothesis

Assume that the statement is true for some arbitrary positive integer \(n=k\), that 3 is a factor of \(k(k+1)(k-1)\). This is the assumption we will use in the next step.

Step 3: Inductive Step

Now, prove that if 3 is a factor of \(k(k+1)(k-1)\), then 3 must be a factor of \((k+1)[(k+1)+1][(k+1)-1] = (k+1)(k+2)(k)\). Rewrite and factorize the expression as \((k+1)k(k+2) = 3 \cdot [(k+1)k(k+2)/3] = (k)(k+1)(k-1) + 3 \cdot (k)(k+1)\). We know that 3 is a factor of \((k)(k+1)(k-1)\) from our induction hypothesis, and \(3 \cdot (k)(k+1)\) is obviously divisible by 3. So 3 must be a factor of \((k+1)(k+2)(k)\). Hence, the proof is complete, the statement holds true for every positive integer \(n\).

Key Concepts

Factorization

Factorization is a key concept in algebra that involves breaking down a complex expression into simpler factors that, when multiplied together, give back the original expression. For example, the expression \(n(n+1)(n-1)\) can be seen as a product of three terms.

• The term \(n\)
• The term \(n+1\)
• The term \(n-1\)

Each term in the product can be considered a factor of the entire expression. Factorization is particularly useful in mathematical induction as it allows us to work with and manipulate these components easily. In our exercise, showing that 3 is a factor of \(n(n+1)(n-1)\) for all positive integers \(n\) requires understanding how these terms interact and contribute to the overall product being divisible by a number like 3.

Positive Integers

A positive integer is any whole number greater than zero. These are sometimes referred to simply as 'natural numbers' when discussing basic arithmetic operations and properties.

• Examples include 1, 2, 3, 4, and so on.
• They do not include fractions, decimals, or negative numbers.

When dealing with problems of mathematical induction, we usually start with the smallest positive integer, often one, as this forms the basis of our proof. Our goal is to establish that a particular property or pattern holds for all positive integers, illustrating its universality within this infinite set.

Base Case

The base case is a crucial step in mathematical induction that involves proving the truth of the given statement for an initial value of the variable, typically the smallest positive integer.

• In this context, we start by checking \(n=1\).
• We verify the expression \(1(1+1)(1-1) = 0\).

Since any number can divide zero, we confirm that 3 is a factor of this expression. Establishing the base case creates the foundation for the inductive step, ensuring our proof is valid from the beginning. If the base case fails, the entire induction process breaks down, which is why it is vital to complete this step accurately.

Inductive Step

The inductive step involves proving that if the assertion is true for some arbitrary positive integer \(k\), then it must also be true for the next integer \(k+1\). This step is often seen as the 'backbone' of mathematical induction.

• Start by assuming the statement holds for \(n = k\).
• The statement says 3 is a factor of \(k(k+1)(k-1)\).

Next, we need to demonstrate that 3 is also a factor of \( (k+1)(k+2)(k) \). By manipulating and rearranging the expression, we rely on the assumption that 3 divides \((k)(k+1)(k-1)\), and show that it logically extends to the next case. Thus, completing the inductive step confirms the statement is true for all successive integers, effectively proving it for the entire set of positive integers.