# Thread: Exponential factorial proof help! (Determine the ones digit of 10!!. Explain....)

1. ## Exponential factorial proof help! (Determine the ones digit of 10!!. Explain....)

I received this problem in class and I am having trouble.

It states:

Define the exponential factorial n!! for positive integers n by 1!!=1 and for n greater than or equal to 1, (n+1)!!= (n+1)^(n!!).
a) Determine the ones digit of 10!!. Explain.
b) Determine the ones digit of 9!!. Explain.
c) Determine the ones digit of 8!!. Explain.
d) Determine, with proof, all digits 0-9 which appear as the ones digit of n!! for at least one positive integer n.

So far I have:
a) 10!!= 3,840 Ones= 0
b) 9!!= 945 Ones=5
c) 8!!= 384 Ones= 4
d) I know that this must be proven by induction. I know that the base case is when n=0. However, I do not know how to continue with the proof.
Thank you for the help in advance.

2. Originally Posted by ljman
I received this problem in class and I am having trouble.

It states:

Define the exponential factorial n!! for positive integers n by 1!!=1 and for n greater than or equal to 1, (n+1)!!= (n+1)^(n!!).
a) Determine the ones digit of 10!!. Explain.
b) Determine the ones digit of 9!!. Explain.
c) Determine the ones digit of 8!!. Explain.
d) Determine, with proof, all digits 0-9 which appear as the ones digit of n!! for at least one positive integer n.

So far I have:
a) 10!!= 3,840 Ones= 0
b) 9!!= 945 Ones=5 ...... how did you get that?! According your definition:

(n+1)!!= (n+1)^(n!!).

9!! = 9^(8!!)

The unit digits of power of 9 are either 1 or 9

c) 8!!= 384 Ones= 4 ...... how did you get that?! Please show work.
d) I know that this must be proven by induction. I know that the base case is when n=0. However, I do not know how to continue with the proof.
Thank you for the help in advance.
.

3. You should probably start by calculating 2!!, 3!!, etc. Looks like you don't yet understand how it works.

4. Originally Posted by lev888
You should probably start by calculating 2!!, 3!!, etc. Looks like you don't yet understand how it works.
I understood that the double factorial of 10 would be all of the even numbers from ten and under. So, the answer for 10!! would be (10 x 8 x 6 x 4x 2)= 3,840.

5. Originally Posted by ljman
I received this problem in class and I am having trouble.

It states:

Define the exponential factorial n!! for positive integers n by 1!!=1 and for n greater than or equal to 1, (n+1)!!= (n+1)^(n!!).
a) Determine the ones digit of 10!!. Explain.
b) Determine the ones digit of 9!!. Explain.
c) Determine the ones digit of 8!!. Explain.
d) Determine, with proof, all digits 0-9 which appear as the ones digit of n!! for at least one positive integer n.

So far I have:
a) 10!!= 3,840 Ones= 0
b) 9!!= 945 Ones=5
c) 8!!= 384 Ones= 4
d) I know that this must be proven by induction. I know that the base case is when n=0. However, I do not know how to continue with the proof.
Thank you for the help in advance.
Your definition of "!!" is incorrect. According to Wolfram:

The double factorial of a positive integer is a generalization of the usual factorial defined by
 (1)

Note that , by definition (Arfken 1985, p. 547)...........................http://mathworld.wolfram.com/DoubleFactorial.html

6. Originally Posted by ljman
I understood that the double factorial of 10 would be all of the even numbers from ten and under. So, the answer for 10!! would be (10 x 8 x 6 x 4x 2)= 3,840.
The definition in the problem statement does not mention odd and even numbers. Which definition are you supposed to use?

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