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Mathematical Induction - Algebra Manipulation

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data
    I am working on a mathematical induction problem, where I need to prove:

    (1 - (-7) ^ (k + 2)) / 4


    2. Relevant equations

    (1 - (-7) ^ (k + 1)) / 4 + 2(-7) ^ (k + 1)

    3. The attempt at a solution

    So I just need to add the two items in section two above. Now I know I need a common denominator (4):

    (1 - (-7) ^ (k + 1)) / 4 + 8(-7) ^ (k + 1) / 4

    Unfortunately, I don't know how to progress any further. I don't know how to add the above together. Could someone please provide a few hints to move me along?
     
    Last edited: Sep 15, 2012
  2. jcsd
  3. Sep 15, 2012 #2
    What exactly are you asked to prove?
     
  4. Sep 16, 2012 #3

    Mentallic

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    You need to prove an expression?
    Can you prove x2?
     
  5. Sep 16, 2012 #4
    I am trying to prove that:
    2 - 2 * 7 + 2 * 7^2 - ... + 2(-7)^n = (1-(-7)^(n+1))/4

    n is a non-negative integer

    I did the basis step

    I'm at the point where I need to prove P(k) -> P(k + 1)
    Therefore, swapping in k+1 for n on the right handside above yields:
    (1 - (-7) ^ (k + 2)) / 4

    Now I need to add 2(-7)^(k+1) to the right side:
    (1 - (-7) ^ (k + 1)) / 4 + 2(-7) ^ (k + 1)

    The issue is that I don't know how to simplify (1 - (-7) ^ (k + 1)) / 4 + 2(-7) ^ (k + 1)
     
  6. Sep 16, 2012 #5

    SammyS

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    Yes, you need to assume P(k) , that is, assume [itex]\displaystyle 2+2(-7)+2(-7)^2+2(-7)^3+\dots+2(-7)^{k}=\frac{1-(-7)^{k+1}}{4}\ .[/itex]

    From that assumption, you need to show P(k+1), which is:

    [itex]\displaystyle 2+2(-7)+2(-7)^2+2(-7)^3+\dots+2(-7)^{k}+2(-7)^{k+1}=\frac{1-(-7)^{k+2}}{4}\ .[/itex]

    Added in Edit:

    Oh, I see that you already know this.


    For [itex]\displaystyle \frac{1-(-7)^{k+1}}{4}+2(-7)^{k+1}=\frac{1-(-7)^{k+2}}{4}\ ,[/itex]

    look at simplifying [itex]\displaystyle \frac{1-(-7)^{k+2}}{4}-\frac{1-(-7)^{k+1}}{4}\ .[/itex]
     
    Last edited: Sep 16, 2012
  7. Sep 16, 2012 #6

    Mentallic

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    My teacher used to always chuck a tantrum whenever we worked with both sides of the equation we were trying to prove. If your teacher is anything like mine and you can only manipulate one side of the equation, then start with the left.

    You have [itex]\displaystyle \frac{1-(-7)^{k+1}}{4}+2(-7)^{k+1}[/itex] and you want to make it equal to [itex]\displaystyle \frac{1-(-7)^{k+2}}{4}[/itex]. Well the first thing you should instantly think of is to make it all one fraction - that is, it should be something / 4.
    After that, you know that both numerators must be equal because the denominators are equal, hence you're already that much closer to the answer.
     
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