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Can someone use induction to solve this please?

  1. Jan 22, 2010 #1
    this is to figure out formula for how many squares you can find in a nxn box.

    12+22+32+.....+N2

    Can someone show steps to how a simplified formula can be found? ( i only know the concept of induction, not how to do it really..)

    it is (n)(n+1)(2n+1)/6 , but how is this accomplished

    thanks a lot, would be great if some work was shown.
     
  2. jcsd
  3. Jan 22, 2010 #2

    berkeman

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    Staff: Mentor

    What is the context of the question? Is this for school work?
     
  4. Jan 23, 2010 #3
  5. Jan 23, 2010 #4
    First show the base case:
    1^2 = 1
    Then assume:
    [tex]1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}[/tex]
    Now add [itex](n+1)^2[/itex] to get:
    [tex]1^2+2^2+\cdots+n^2+(n+1)^2 = \frac{n(n+1)(2n+1)}{6}+(n+1)^2[/tex]
    Now your task is do manipulate the right hand side to get:
    [tex]\frac{n(n+1)(2n+1)}{6}+(n+1)^2 = \frac{(n+1)(n+2)(2n+3)}{6}[/tex]
    which would prove the induction hypothesis that if the formula is true for n, then it's true for n+1.

    If you do this you show the theorem true for n=1 and therefore also by n=2, and therefore also for n=3, and therefore also for n=4, .... So by induction you have shown it true for all positive n.
     
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