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Math Help - Need help proving mathematical induction

  1. #1
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    Need help proving mathematical induction

    Prove that 1(1!) + 2(2!) + ... + n(n!) = (n+1)! - 1, for n >= 1

    Here's what I have:

    Base case: n = 1.
    s(n) = 1(1!) + 2(2!) + ... + n(n!) = (n+1)! - 1
    Therefore, (1+1)! - 1 = 1
    Correct...

    Inductive step:
    For n+1,
     n(n!) = (n+1)*(n+1)!

    s(n+1) = s(n) + (n+1)*(n+1)!
    or...
    s(n+1) = (n+1)! - 1 + (n+1)*(n+1)!

    This is where I am stuck. What next?
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  2. #2
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    Quote Originally Posted by kalel918 View Post
    Prove that 1(1!) + 2(2!) + ... + n(n!) = (n+1)! - 1, for n >= 1

    Here's what I have:

    Base case: n = 1.
    s(n) = 1(1!) + 2(2!) + ... + n(n!) = (n+1)! - 1
    Therefore, (1+1)! - 1 = 1
    Correct...

    Inductive step:
    For n+1,
     n(n!) = (n+1)*(n+1)!

    s(n+1) = s(n) + (n+1)*(n+1)!
    or...
    s(n+1) = (n+1)! - 1 + (n+1)*(n+1)!

    This is where I am stuck. What next?
    (n+1)! - 1 + (n+1)*(n+1)! = (n+1)! (1 + [n+1]) - 1 = (n+1)!(n+2) - 1 = (n+2)! - 1
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    (n+1)! - 1 + (n+1)*(n+1)! = (n+1)! (1 + [n+1]) - 1 = (n+1)!(n+2) - 1 = (n+2)! - 1
    Thank you so much for your help. I cannot thank you enough.

    However, I am still confused as to how you reached the solution.

    (n+1)! (1 + [n+1]) - 1

    From what I see here, you factored out (n+1)!, correct? I follow at this point...

     = (n+1)!(n+2) - 1 = (n+2)! - 1

    Okay, you lost me here. Can you please clarify this once more?
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  4. #4
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    Quote Originally Posted by kalel918 View Post
    Thank you so much for your help. I cannot thank you enough.

    However, I am still confused as to how you reached the solution.

    (n+1)! (1 + [n+1]) - 1

    From what I see here, you factored out (n+1)!, correct? I follow at this point...

     = (n+1)!(n+2) - 1 = (n+2)! - 1

    Okay, you lost me here. Can you please clarify this once more?
    (n + 2)! = (n+2) {\color{red}(n+1)n(n-1) ..... 1} = (n+2) {\color{red}(n+1)!}.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    (n + 2)! = (n+2) {\color{red}(n+1)n(n-1) ..... 1} = (n+2) {\color{red}(n+1)!}.
    Wow, it just clicked for me. Thank you so much for your help.
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