Question

Show that (a) \(\frac{n !}{(n-1) !}=n\), (b) \(\frac{(n+1) !}{(n-1) !}=n(n+1)\) (c) \(\frac{n !}{(n+1) !}=\frac{1}{n+1}\).

Short answer

Based on the step-by-step solution provided, please answer the following question: **Question:** Prove that \(\frac{n!}{(n-1)!} = n\). **Answer:** Given the factorial expression \(\frac{n!}{(n-1)!}\), we can expand it using the definition of a factorial: \(\frac{n!}{(n-1)!} = \frac{n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1}{(n-1) \cdot (n-2) \cdots 2 \cdot 1}\) By cancelling out the common terms \((n-1)\), \((n-2)\), \(\cdots\), \(2\), and \(1\) in the numerator and the denominator, we are left with: \(n\) Hence, we have proven that \(\frac{n!}{(n-1)!} = n\).

Step-by-step solution

Proof of Expression (a): \(\frac{n !}{(n-1) !}=n\)

Recall the definition of a factorial: \(n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1\). Now, let's write the expression and expand it using the definition: \(\frac{n!}{(n-1)!} = \frac{n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1}{(n-1) \cdot (n-2) \cdots 2 \cdot 1}\) We can see that \((n-1)\), \((n-2)\), \(\cdots\), \(2\), and \(1\) all cancel out in the numerator and the denominator, leaving: \(n\) So, we have proven that \(\frac{n !}{(n-1) !}=n\).

Proof of Expression (b): \(\frac{(n+1) !}{(n-1) !}=n(n+1)\)

Expand the expression using the definition of a factorial: \(\frac{(n+1)!}{(n-1)!} = \frac{(n+1) \cdot n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1}{(n-1) \cdot (n-2) \cdots 2 \cdot 1}\) Here, we can cancel out \((n-1)\), \((n-2)\), \(\cdots\), \(2\), and \(1\) in the numerator and the denominator, leaving: \((n+1) \cdot n\) This simplifies to \(n(n+1)\), so we have proven that \(\frac{(n+1) !}{(n-1) !}=n(n+1)\).

Proof of Expression (c): \(\frac{n !}{(n+1) !}=\frac{1}{n+1}\)

Expand the expression using the definition of a factorial: \(\frac{n!}{(n+1)!} = \frac{n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1}{(n+1) \cdot n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1}\) Now, we can cancel out \(n\), \((n-1)\), \((n-2)\), \(\cdots\), \(2\), and \(1\) in the numerator and the denominator, leaving: \(\frac{1}{n+1}\) So, we have proven that \(\frac{n !}{(n+1) !}=\frac{1}{n+1}\).

Key concepts

Mathematical Proofs

Simply put, a mathematical proof is a logical argument that demonstrates the truth of a given statement. It's the gold standard for verifying that something is definitely true in the world of mathematics. These proofs can come in various forms, such as direct proofs, contradiction, induction, and more. What's beautiful about proofs is that once something is proven, it's considered reliable and universally accepted across the mathematical community.

In the exercise, we tackled the proofs of certain factorial expressions. This involved expanding factorials, canceling terms, and arriving at an elegant, simplified result. Understanding proofs not only allows students to verify these expressions but also builds a foundation for more complex mathematical reasoning. Think of each proof as a building block in a large and infinitely interesting structure that is mathematics.

Factorial Definition

Before diving into factorial expressions, it's essential to understand what a factorial is. The factorial of a non-negative integer 'n', denoted by 'n!', is the product of all positive integers less than or equal to 'n'. In more familiar terms, it's like saying we're throwing a party where 'n' friends are coming, and we want to see how many different ways they can line up to take a group photo.

It's worth noting that the factorial function grows very rapidly with every increase in 'n'. This exponential growth is crucial to comprehend since the factorial plays a vital role in areas like combinatorics, probability, and algebra. For example, the formulas involving permutations and combinations heavily rely on factorials to determine the number of possible arrangements or selections.

Algebraic Manipulation

Algebraic manipulation is the art of transforming mathematical expressions into a desired or simpler form using a set of rules and operations. It's like having a toolbox where you can take apart an expression and put it back together in a way that's easier to understand or use. These tools include operations like addition, subtraction, multiplication, division, and factoring; they allow us to tease out patterns and simplify complex expressions.

In our factorial expressions exercise, we applied these skills of algebraic manipulation to cancel out terms and simplify the expressions. Mastering algebraic manipulation makes it much more comfortable to work with equations, solve for variables, and understand the relationships between different mathematical elements.