Question

Prove the double factorial identities: $$ \begin{gathered} (2 n) ! !=2^{n} n ! \\ (2 n-1) ! !=\frac{(2 n) !}{2^{n} n !} \end{gathered} $$

Short answer

Both the double factorial identities are true, for even numbers \((2n)! = 2^n * n!\) and for odd numbers \((2n-1)! = \frac{(2n)!}{2^n * n!}\). This has been proven by validating that both sides of the equations produce the same sequential product of numbers.

Step-by-step solution

Prove for even numbers

First, prove the double factorial identity for even numbers. Double factorial is only calculated for nonnegative integers. For an even number \(2n\), the expression \((2n)!\) shows a double factorial and is evaluated by the formula \((2n)! = 2^n * n!\). It means if we multiply every second integer from 1 up to \(2n\), it will equal \(2^n * n!\). Let's prove this by showing that both sides of the equation produce the same sequence of multiplied numbers. The left hand side \((2n)!\) calculates as \(1*2*3*4...*(2n)\), and the right hand side calculates as \(2^n * n!\), which is \(2^1*2^2*2^3...*2^n * 1*2*3...*n\). If we simplify, it gives the same sequence of numbers multiplied. Thus, this identity holds for even numbers.

Prove for odd numbers

Now, prove the double factorial identity for odd numbers. The expression for an odd number is \((2n-1)!\) and it should be equal to \(\frac{(2n)!}{2^n*n!}\). Starting from the left side which computes as \(1*2*3...*(2n-1)\), meanwhile the right side calculates as \(\frac{2^1*2^2*2^3...*2^n * 1*2*3...*n * 1*2*3...*2n}{2^1*2^2*2^3...*2^n * 1*2*3...*n}\). If we simplify this, it gives the same sequence of numbers multiplied from 1 up to \(2n-1\), hence, thus this identity holds for odd numbers.

Key concepts

Mathematical Proof

The concept of a mathematical proof is fundamental in understanding how mathematicians establish the truth of a statement beyond any doubt. A proof is a logical argument composed of a series of statements, each deduced from the previous one or accepted principles known as axioms. To prove an identity or equation, we must show that both sides of the equation are equivalent under all relevant conditions. This usually involves manipulating expressions algebraically or invoking other established theorems to substantiate the argument. When it comes to factorial identities, the proofs ensure that the definitions hold and can be relied upon for both even and odd numbers. Proofs can be intuitive, but they require rigorous verification to be accepted within the mathematical community.

Factorial

The factorial, denoted by the exclamation mark (!), plays a critical role in combinatorics, probability, and various fields of mathematics. For any non-negative integer n, the factorial n! is the product of all positive integers less than or equal to n. Formally, it's defined as:
\[ n! = n \times (n-1) \times (n-2) \times \text{...} \times 2 \times 1 \]
For example, 5! would be 5 \times 4 \times 3 \times 2 \times 1 = 120. In the context of our exercise, a double factorial identity involves two factorial operations and the relationship between them, particularly on even or odd sequences of numbers.

Even and Odd Numbers

Even and odd numbers have distinct properties that are often leveraged in mathematical proofs and theories. An even number is any integer that can be exactly divided by 2, while an odd number has a remainder of 1 when divided by 2. In algebraic equations and identities, the behavior of expressions can significantly differ based on the parity of the numbers involved. For instance, multiplying a series of consecutive even or odd numbers will yield different products, and the mathematical patterns that emerge from these sequences are what underlie the double factorial identities explored in the exercise. Recognizing whether a number is even or odd is crucial for correctly applying the respective identity and understanding the underlying structure of these numerical patterns.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. These are powerful tools in simplifying and solving algebraic expressions and are used extensively in various mathematical proofs. Examples include the square of a binomial ((a + b)^2 = a^2 + 2ab + b^2) and the difference of squares (a^2 - b^2 = (a + b)(a - b)). In the context of the exercise on double factorial identities, understanding algebraic manipulation is essential. By arranging and simplifying the expressions, we can demonstrate the validity of the double factorial identities. Such identities are themselves algebraic truths that can help simplify complex problems, especially in combinatorics and number theory.