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

[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.

 

[Deleted member 4993]

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.

.

 

[lev888]

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

 

[ljman]

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.

 

[Deleted member 4993]

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

 

[lev888]

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?