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

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    Post 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.

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    Quote Originally Posted by ljman View Post
    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.
    .
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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

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    Quote Originally Posted by lev888 View Post
    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.

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    Quote Originally Posted by ljman View Post
    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
    Last edited by Subhotosh Khan; 03-07-2018 at 09:10 AM.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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    Quote Originally Posted by ljman View Post
    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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