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.
To find out the product of even numbers think of the factorial. [tex]n!=1.2.3.4...(n-1)n[/tex] So you want to find the product [tex]f(n)=2.4.6.8...(2(n-1))(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(n-1))(2n)[/tex] Next you want to find the product such that [tex]f(n)=1.3.5.7...(2(n-1)-1)(2n-1) [/tex] Its similar... divide each one by 2 and [tex]2^{-n}f(n)=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n-3}{2}\frac{n-1}{2} [/tex] [tex]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}) [/tex] [tex](n-\frac{1}{2})!=(n-\frac{1}{2})(n-1-\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...(2n-1)[/tex] So now you have the formula for both things