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Formula for even and odd number multiplication 
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#1
Apr1210, 08:48 AM

P: 66

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
Apr1210, 10:51 AM

P: 463

To find out the product of even numbers think of the factorial.
[tex]n!=1.2.3.4...(n1)n[/tex] So you want to find the product [tex]f(n)=2.4.6.8...(2(n1))(2n) [/tex] Notice that this is just the normal factorial function but each number has been multiplied by 2. So it is [tex] 2^n [/tex] bigger. So it is [tex]f(n)=2^nn! = 2.4.6.8...(2(n1))(2n)[/tex] Next you want to find the product such that [tex]f(n)=1.3.5.7...(2(n1)1)(2n1) [/tex] Its similar... divide each one by 2 and [tex]2^{n}f(n)=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n3}{2}\frac{n1}{2} [/tex] [tex]2^{n}f(n)=(1\frac{1}{2})(3\frac{1}{2})(5\frac{1}{2})...((n1)\frac{1}{2})(n\frac{1}{2}) [/tex] [tex](n\frac{1}{2})!=(n\frac{1}{2})(n1\frac{1}{2})...\frac{5}{2}\frac{3}{2}\frac{1}{2}\sqrt{\pi}[/tex] To work out why this is so look at the definition for the Gamma function. [tex]\frac{2^n}{\sqrt{\pi}}(n\frac{1}{2})! = 1.3.5.7...(2n1)[/tex] So now you have the formula for both things 


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