Formula for even and odd number multiplication

[gaobo9109]

Can anyone tell me what's the formula for even and odd number multiplication.
For example, what would be the product for 2 x 4 x 6 x 8 x 10 ... x 100 and what would be the product for 1 x 3 x 5 x 7 x 9 x 12 ... x 99? I am trying to solve a problem which asks me to prove that 1/2 x 3/4 x 5/6 ... x 99/100 < 1/10. And i think finding the formula is key to solving this question.

 

[Gregg]

To find out the product of even numbers think of the factorial.

$$n!=1.2.3.4...(n-1)n$$

So you want to find the product

$$f(n)=2.4.6.8...(2(n-1))(2n)$$

Notice that this is just the normal factorial function but each number has been multiplied by 2. So it is $$2^n$$ bigger.

So it is

$$f(n)=2^nn! = 2.4.6.8...(2(n-1))(2n)$$

Next you want to find the product such that

$$f(n)=1.3.5.7...(2(n-1)-1)(2n-1)$$

Its similar... divide each one by 2 and

$$2^{-n}f(n)=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n-3}{2}\frac{n-1}{2}$$

$$2^{-n}f(n)=(1-\frac{1}{2})(3-\frac{1}{2})(5-\frac{1}{2})...((n-1)-\frac{1}{2})(n-\frac{1}{2})$$

$$(n-\frac{1}{2})!=(n-\frac{1}{2})(n-1-\frac{1}{2})...\frac{5}{2}\frac{3}{2}\frac{1}{2}\sqrt{\pi}$$

To work out why this is so look at the definition for the Gamma function.

$$\frac{2^n}{\sqrt{\pi}}(n-\frac{1}{2})! = 1.3.5.7...(2n-1)$$

So now you have the formula for both things