Problem 27 - College Algebra

Question

In Exercises \(11-30,\) 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

By the Principle of Mathematical Induction, it is proved that 3 is indeed a factor of \(n(n+1)(n-1)\) for every positive integer \(n\).

Step 1: Prove the Base Case

To establish the base case, substitute \(n=1\) into the expression. This gives \(1(1+1)(1-1) = 0\), and indeed, 3 is a factor of 0, as any number is a factor of 0. Hence, the hypothesis is valid for \(n=1\).

Step 2: Assumption

Assume the statement is true for some \(n=k\). Thus, 3 is a factor of \(k(k+1)(k-1)\). Let \(P(k)\) denote this statement. This means that \(k(k+1)(k-1)\) can be written as \(3m\) for some integer \(m\), because it is a multiple of 3.

Step 3: Induction Step

We need to show that the statement is true for \(n=k+1\). Thus, we need to prove that 3 is a factor of \((k+1)((k+1)+1)((k+1)-1)\). Simplifying this gives \(3(k^2 +k)\). Since we assumed that \(k(k+1)(k-1)\) is a multiple of 3, we can substitute \(k+1 = m+1\), and that simplifies to \(3m(m+2)\), which is clearly divisible by 3. Therefore, the statement is true for \(n=k+1\), if it is true for \(n=k\).

Step 4: Conclusion

By the Principle of Mathematical Induction, the statement is true for all positive integers \(n\). The base case \(n=1\) is true, and if the statement is true for an arbitrary positive integer \(n=k\), it is also true for \(n=k+1\). Therefore, the statement holds for all positive integers \(n\).

Key Concepts

Factorization

Factorization is an essential mathematical process that involves breaking down expressions, numbers, or other mathematical objects into products of simpler or more basic components. In algebra, this often means expressing a polynomial as a product of its factors. Factorization is incredibly useful because it can simplify complex problems or make certain properties more apparent.
For example, if you are given a polynomial expression like \(n(n+1)(n-1)\), it can be insightful to recognize it as a product of three consecutive numbers, making it easier to prove statements about divisibility, such as determining common factors like 3.

• Recognizing a factored form helps in simplifying calculations.
• It allows for easier handling of algebraic expressions, especially in proofs.
• Factorization has applications in solving equations, both linear and quadratic.
• Helps in identifying properties like divisibility, as shown in the exercise.

Positive Integers

Positive integers are the numbers that serve as the foundation of arithmetic and algebra. They are numbers greater than zero and do not include any fractions or decimals. Not only are they essential for counting, but they also play a crucial role in various mathematical theorems and proofs.
When working with positive integers in mathematical induction, these integers start from 1 (e.g., 1, 2, 3, 4, ...). Induction often relies on establishing truth for these integers to cover an entire set. As seen in the exercise, showing that a property holds for every positive integer \(n\) ensures the statement applies universally.

• Positive integers are used in counting sequences and sets.
• They are critical in proving theorems involving properties like divisibility.
• In mathematical induction, they are part of establishing the basis and inductive steps.

Algebraic Proof

An algebraic proof involves using algebraic techniques and principles to demonstrate the truth of a mathematical statement. Such proofs apply logical reasoning through algebraic manipulation to verify assertions regarding numbers and expressions.
In the context of mathematical induction, an algebraic proof is employed to ensure that the statement applies to all positive integers \(n\), starting with showing the truth for a base case and then proving it holds for the successive integers. This is achieved by assuming it is valid for some arbitrary integer \(n=k\) and proving it for \(n=k+1\).

• Algebraic proofs use logical steps to simplify and validate statements.
• They often involve recognizing patterns and utilizing known algebraic identities.
• Provide a structured framework for proving statements across infinite sets.
• Critical part of understanding and verifying divisibility, as highlighted in exercises utilizing induction.