how does 2n! decrease

To understand how 2n! decreases, let's first discuss what the notation "n!" means. n! (read as "n factorial") represents the factorial of a non-negative integer, which is the product of all positive integers from 1 to n.

For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Now, let's consider 2n!. This expression represents the factorial of 2n, which is the product of all positive integers from 1 to 2n.

As n increases, the value of 2n! will also increase because the factorial includes all the positive integers up to 2n. However, if we specifically look at the ratio between (2n)! and ((2n)-1)!, we can observe that it will gradually decrease.

For example, let's compare the values of 6! and 5!.

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

5! = 5 x 4 x 3 x 2 x 1 = 120

Here, we can see that 6! is greater than 5! by a factor of 6. In general, we can notice that the ratio between (2n)! and ((2n)-1)! is always equal to 2n.

So, as n increases, the ratio between (2n)! and ((2n)-1)! becomes larger because 2n is included in the factorial. However, if we look at the actual value of 2n!, it will still increase since it includes all positive integers up to 2n.