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Make a conjecture about the sum? Confused on what they want exactly

  1. Oct 2, 2006 #1
    Hello everyone,

    I'm confused on the directions. It says, Evalute the sum, for n = 1, 2, 3, 4, and 5. Make a conjecture about a formula for this sume for general n, and prove your conjecture by mathematical induction.

    This is the sum and my work:
    [​IMG]


    I'm having a hard time figuring out that formula, and what exactly is a conjecture? In the other readings and this one they don't say anything about a conjecture.

    I was thinking that maybe they just want me to place an n where the k is?
    like n/(n+1)! ?

    Thanks!
     
    Last edited: Oct 2, 2006
  2. jcsd
  3. Oct 2, 2006 #2

    StatusX

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    Why don't you actually add up the fractions? You should see a pattern emerge.
     
  4. Oct 2, 2006 #3
    I kind of see a pattern at the very last 2 terms,
    it goes from 119/120 to 719/720, the numerator is incremented by 600 and the denominator also by 600.
    But before that it goes
    n = 1
    1/2
    n = 2
    5/6
    n = 3
    23/24
    n = 4
    119/120
    n = 5
    719/120
    n = 6

    if i keep going i get
    n = 7
    2519/2520

    Oo i see it now hah...

    The denominator is +1 the numerator
    but if i use n that won't work for all cases.
    n/n+1, n >= 1

    Is there a technique to approach conjectures? I see the pattern but i don't see how n is producing the numbers.
     
  5. Oct 2, 2006 #4

    StatusX

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    The denominators are 1,2,6,24,120,... Do you recognize these?
     
  6. Oct 2, 2006 #5
    I know they are factorials, like
    1! = 1
    2! = 1*2 = 2
    3! = 1*2*3 = 6
    4! = 1*2*3*4 = 24
    5! = 1*2*3*4*5 = 120

    which is 1/n!
     
  7. Oct 2, 2006 #6
    I think i got it, am i allowed to write the following?

    Let n be an integer, where n >= 2 such that Sumnation k = 2 to n (n!-1)/n!

    THis also doesn't work i just saw, because i can't change the orginal sum, which is letting k = 1, i can't just change it can i?

    Is this what a conjecture is? I'm guessing the pattern of the output and I must prove the right and left are equal?

    [​IMG]
     
    Last edited: Oct 2, 2006
  8. Oct 2, 2006 #7

    StatusX

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    Well then just use (n+1)! instead of n!. By "conjecture" they mean find a formula that seems right (works for low numbers). Then you need to prove the formula is correct using induction.
     
  9. Oct 2, 2006 #8
    Ahh yes! it works, thanks a ton Status! :smile:
     
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