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Properties of integers:math induction

  1. Nov 17, 2008 #1
    1. The problem statement, all variables and given/known data
    consider the following four equations:
    1) 1=1
    2) 2 + 3 + 4 = 1 + 8
    3) 5 + 6 + 7 + 8 + 9 = 8 + 27
    4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64

    conjecture the general formula suggested by these four equations and prove
    your conjecture


    2. Relevant equations



    3. The attempt at a solution

    I know the left side has n + (n+1) numbers,
    and the right side has the last number occuring the in next equation,
    i also see that the numbers on the right side come from an k^3 number.
    but how would i write this in a general formula which would be sufficient to
    satisfy all the equations? because to me it seems like a recursive definition but
    i don't quite clearly understand how i would write out an answer for this....
    and also what exactly am i supposed to write down to solve this??
     
  2. jcsd
  3. Nov 17, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Notice also that the last number in the sum on the left is always a square, say, n2 and the last number on the right is n3. Use that as your base.
    It looks to me like the first number on the left is (n-1)2+ 1 so the sum on the left is from (n-1)2+ 1 to n2.
     
  4. Nov 17, 2008 #3
    so on the left side........ could i use sigma notation to verify the range and the addition of the numbers?? and on the right side ....... i can see that it's just like
    1^3,
    1^3 + 2^3 ,
    2^3 + 3^3
    3^3 + 4^3
    how would i show this in terms of just using n ?
    n + (n)^3 ?

    the problem with this is that it will only apply to one specific instant of the equation,
    or am i wrong??
     
  5. Nov 17, 2008 #4

    HallsofIvy

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    Staff Emeritus
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    On both left and right side you are going to have to specify what "n" is!
     
  6. Feb 25, 2010 #5
    I guess the conjecture is
    n^2 - 2 (n - 1) +....+ n^2 = (n-1)^3 + n^3
    and it holds by trivial processing of the two sides
     
  7. Feb 25, 2010 #6

    Mark44

    Staff: Mentor

    Does your guess work for the given equations?
    There's a lot more to this than "trivial processing." Note the title of the thread.
     
  8. Feb 25, 2010 #7
    Yes it does
    in the formula n^2 - 2 (n - 1) +......+ n^2 = (n-1)^3 + n^3
    1) 1=1
    put n = 1
    2) 2 + 3 + 4 = 1 + 8
    put n = 2
    3) 5 + 6 + 7 + 8 + 9 = 8 + 27
    put n = 3
    4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64
    put n = 4
    Yep! I didn't get to prove it by induction.
     
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