Problem 24 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The simplified form of the ratio \(\frac{(2 n+2) !}{(2 n) !}\) is \((2n+2)(2n+1)\).

Step 1: Recognize Larger Factorial

Recognize that the factorial of a larger number, in this case, \(2n+2\), can be expressed in terms of smaller numbers. So, \( (2n+2)! = (2n+2)(2n+1)(2n)!\).

Step 2: Simplify the Ratio

Now replace \( (2n+2)! \) in the original problem with the equivalent form formulated in the first step. Therefore, \(\frac{(2 n+2) !}{(2 n) !}\) becomes \(\frac{(2n+2)(2n+1)(2n)!}{(2 n) !}\)

Step 3: Cancel Out Common Factors

The common factors in both the numerator and denominator of the rational expression - in this case, \( (2n)! \) - can be canceled out. So, \(\frac{(2n+2)(2n+1)(2n)!}{(2 n) !}\) simplifies as \((2n+2)(2n+1)\).

Key Concepts

Ratio Simplification

Simplifying ratios involves reducing the expression to its simplest form by eliminating any common factors shared between the numerator and the denominator. Using a straightforward method such as cancellation, you can identify and eliminate these shared factors.

This can make complex expressions easier to work with.

• First, express both parts of the ratio in terms of multiplication or division.
• Next, find any common factors in the numerator and denominator.
• Finally, cancel out these common factors to obtain a simpler expression.

In the case of simplifying the ratio of factorials such as in the original exercise, the key step is recognizing those common factors that occur due to the properties of factorials. This can drastically simplify even complex mathematical expressions by reducing them to a combination of simple multipliers.

Factorial Notation

Factorial notation, symbolized as an exclamation mark (e.g., \(!\) ) next to a number or variable, represents the product of all positive integers up to that number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Factorial calculations are prevalent in permutations, combinations and various areas within mathematical analysis.

• Factorial expressions grow very quickly as the number increases, which means that the base number plays a crucial role in simplification.
• Always break down the factorial into a product of its sequence, which can reveal common factors easily.
• In expressions like \((2n+2)!\), utilizing the properties of factorials helps reduce complexity by expanding and simplifying the sequence.

Factorials provide a compact notation for expressing large multiplication sequences, and understanding how to manipulate these sequences can greatly aid in solving problems involving ratios of factorials.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the validity of mathematical statements. They are crucial for verifying formulas and equations, ensuring that they hold true in all relevant cases.

There are several types of mathematical proofs, but most involve logical reasoning, algebraic manipulation, and clear presentation.

• Recognize the requirement of supporting each step with valid reasoning and known mathematical principles.
• In problems like simplifying the ratio of factorials, the proof consists of showing each change respects algebraic rules and properties of factorials.
• Visualizing the sequence of transformations and cancellations is often a helpful method to verify the proof step-by-step.

Remember that a proof isn't just about reaching the correct answer; it's about providing a methodical explanation that others can follow and understand. This ensures clarity and logical coherence in mathematical problem-solving.