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Rewriting (1 + 2 + + [k + 1])^2

  1. Dec 24, 2013 #1
    This is from Spivak Calculus 4th ed.

    Why are these two formulas equal?

    Code (Text):
    (1 + 2 + ... + [k + 1])[SUP]2[/SUP] = (1 + 2 + ... + k)[SUP]2[/SUP] + 2(1 + ... + k)(k + 1) + (k + 1)[SUP]2[/SUP]
     
    I'm especially confused about the right hand side. Where did 2(1 + ... + k)(k + 1) come from?
     
  2. jcsd
  3. Dec 24, 2013 #2

    jedishrfu

    Staff: Mentor


    Didn't they just group them into two terms?

    (x + (k + 1))^2 = x^2 + 2*x*(k+1) + (k+1)^2

    where x = 1 + 2 ... + k
     
  4. Dec 24, 2013 #3

    LCKurtz

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    Science Advisor
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    Gold Member

    Try it with k = 3. Start with$$
    (1+2+3+4)(1+2+3+4) = [(1+2+3)+4)][(1+2+3)+4]$$Expand as a binomial without simplifying and see what happens. That might give you an idea.

    [Edit] I didn't see the other post while I was typing.
     
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