Product of even numbers divided by product of odd numbers

Number Theory Level pending

Suppose you multiply even numbers from 1 to 10 -> 2 x 4 x 6 x 8 x 10

and divide that product by the product of odd numbers from 1 to 10 -> 1 x 3 x 5 x 7 x 9

You get a finite real number which is less than 5.

Now let us extend this operation to a higher number n, where the numerator is 2 x 4 x 6 x .... x 2n ............................(1)

and the denominator is 1 x 3 x 5 x 7 x ... x (2n-1) ...............(2)

Dividing (1) by (2) gives you the result.

Let us call this Res(n)

Show how you arrive at the approximate formula for Res(n)

Find an approximate value for this result for n = 50 and n = 500

Let P = Res(500)/Res(50)

What is the value of P correct to two decimal places.

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