Problem 22 - Essential Calculus

Question

Simplify the ratio of factorials. \(\frac{(n+2) !}{n !}\)

Step-by-Step Solution

Answer

The simplified form of the ratio of factorials is \( (n+2) \cdot (n+1) \).

Step 1: Understanding Factorials

The factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\). It can be denoted as \(n!\). Therefore, \(n! = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1\). Similarly, \((n+2)!\) is the product of all positive integers less than or equal to \(n+2\). Thus, \((n+2)! = (n+2) \cdot (n+1) \cdot n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1\).

Step 2: Breaking Down the Expression

To simplify the expression, substitute the definitions of factorials into the fractions: \(\frac{(n+2)!}{n!} = \frac{ ((n+2) \cdot (n+1) \cdot n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1)} {n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1}\).

Step 3: Simplifying the Ratio

It can be observed that \(n!\) is part of \((n+2)!\). The similar parts cancel each other out. This leaves: \( \frac{(n+2)!}{n!} = (n+2) \cdot (n+1)\).

Key Concepts

Simplifying Expressions

Expressions with factorials can seem complex at first. However, simplifying them often becomes straightforward once you understand the basics. Simplifying expressions means making them as simple as possible. This involves reducing them to a form where calculations or interpretations are easier.

When dealing with factorials, a major goal is to cancel out common terms. This reduces the expression, stripping it down to more fundamental components. Let's consider the factorial division \( \frac{(n+2)!}{n!} \).

The key move is identifying similar terms within the numerator and the denominator. Since \( n! \) appears entirely in both the \((n+2)!\) numerator, it can be cancelled. This simplifies the expression drastically.

Remember, simplifying expressions is about recognizing patterns and repeating parts. Reducing what appears complex, to its simplest form.

Ratio of Factorials

A ratio of factorials is essentially a type of division where the top part (numerator) and the bottom part (denominator) are both factorials. This concept is significant in many mathematical fields, including combinatorics and probability.

When you have a factorial in the denominator, a strategy is to factor out this full portion from the numerator. In our example \( \frac{(n+2)!}{n!} \), \((n+2)!\) contains all terms of \(n!\).

• The numerator \((n+2)!\) expands to \((n+2) \cdot (n+1) \cdot n!\).
• The denominator is simply \(n!\).

By discarding the common \(n!\), the simplified ratio becomes \((n+2) \cdot (n+1)\).

Hence, the understanding of ratios of factorials is crucial in simplification. It not only brings elegance but also aids in solving complex mathematical problems.

Mathematical Notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. It is an essential language for expressing mathematical concepts clearly and concisely.

Factorials are one such notation. Represented by an exclamation mark (![n]! for example), it signifies the product of all integers from 1 to \(n\).

• \(n!\) means multiplying all positive integers up to \(n\).
• \((n+2)!\) would include numbers up to \(n+2\).

In simplification, knowing these notations helps recognize and cancel terms efficiently, like in the expression \(\frac{(n+2)!}{n!}\). Recognizing factorial patterns paves the way for simpler computation.

Mathematical notation simplifies complex ideas into understandable segments. Adequate understanding translates into fluency in mathematical problem-solving.