Formula for even and odd number multiplication

1. gaobo9109

69
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.

2. Gregg

463
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

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