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Need help with Simplification step in Inductive Proof

  1. Nov 23, 2015 #1
    1. The problem statement, all variables and given/known data

    Let P (n) be the statement that 1^3 + 2^3 + · · · + n^3 = (n(n + 1)/2)^2 for the positive integer n. Prove inductively.

    2. Relevant equations


    3. The attempt at a solution

    I am skipping a few steps...I just need help here:

    1/4K^2(k + 1)^2 + (k + 1)^3

    Since I have access to the solution, the next step is this:

    1/4(k+1)^2 [K^2 + 4 (k + 1)]

    I am confused at how this is gotten to. I appreciate the help.

    Thanks.
     
  2. jcsd
  3. Nov 23, 2015 #2

    Mark44

    Staff: Mentor

    Yes, you are skipping some steps.
    The first term above is from ##1^3 + 2^3 + \dots + k^3## and the other term above is from adding ##(k + 1)^3## in your induction step.
    In the two terms, do you notice that there is a common factor?
     
  4. Nov 23, 2015 #3
    Ahhh, yes. (k+1).

    For some reason, I was trying to factor out k, k^2, etc.

    So, from factoring out, I am getting:

    1/4(k+1)^2 [k^2 + (k+1)]

    I'm still a little confused as to where the 4, in front of the (k + 1) in the brackets is coming from. I assume it's because we are factoring out a 1/4, so we're just multiplying it by 4, to make it equal to one.
     
  5. Nov 24, 2015 #4

    Mark44

    Staff: Mentor

    The part you left out is fouling you up.
    The induction hypothesis is:
    ##1^3 + 2^3 + 3^3 + \dots + k^3 = \frac{k^2(k + 1)^2}{4}##
    Now, work from the induction step:
    ##1^3 + 2^3 + 3^3 + \dots + k^3 + (k + 1)^3 = \frac{k^2(k + 1)^2}{4} + (k + 1)^3##

    What happens when you factor out ##(k + 1)^2## from the right side?
     
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