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A Challenge Problem in Techniques Of Problem Solving

  1. Jun 18, 2010 #1

    AK2

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    I just need a hint to solve the problem. The method used is illustrated in the example problem as follows:

    I am supposed to imitate the method used in the example problem to solve the challenge problem in the book as follows

    Imitate the method used in the last problem to find a formula for the sum
    12 + 22 + 32 + ... + k2

    when k is a positive integer.

    I have tried various things like (a+1)4 - a4, (a+1)3 - a3, triangle numbers, odd numbers, but I havent been able to solve it. I just need a hint. This is not a homework problem. This is self study.
     
  2. jcsd
  3. Jun 18, 2010 #2
    Use

    [tex] (k+1)^3 - k^3 = 3k^2+3k+1[/tex]

    and do summation on both side. By telescoping method, only 2 terms left in the left hand side. You need to do a little manipulation on the right hand side though.
     
  4. Jun 18, 2010 #3
    Why not like the Contiguous Numbers: for example,P=1+2+3+4+5....+100 and the answer of P is (1+100)*100/2 , so the S=(1+k)*k/2
     
  5. Jun 18, 2010 #4

    AK2

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    Thanks for the hint
    I got my answer for S

    S = (2k3+3k2+k)/6
     
  6. Jun 18, 2010 #5
    You are right. You may want to do some factorization also.

    [tex] S = \frac{ n(n+1)(2n+1)}{6}[/tex]
     
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