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Question: Proof by Induction Watch

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    Hi everyone, a friend of mine said if I was having any problems with maths that I should come here and I'll be able to get help.

    I understand how to do proof by induction. However, I was watching a tutorial video by exam solutions, if you skip to 7:18 and pause, you'll see the steps he took to get it in the form they required. The problem, I'm having is I don't understand how he's doing it. I normally find myself expanding out everything and I usually end up having to factorise quintic equations & cubic equations and etc... Although my way always gives me the correct answer, it takes a very long time. So could someone help me understand what he does please
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    Here's the video http://www.examsolutions.net/maths-r...tutorial-1.php
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    It wont let me link the video one sec guys
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    Guys if you copy and paste this link, I'm sure you'll find the video
    Mathematical Induction - Sum of series for r cubed : ExamSolutions
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    (Original post by NinCheng)
    Guys if you copy and paste this link, I'm sure you'll find the video
    Mathematical Induction - Sum of series for r cubed : ExamSolutions
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    7:18 on that video, i labelled the lines 1 to 5, which one aren't you understanding?
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    (Original post by DylanJ42)
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    7:18 on that video, i labelled the lines 1 to 5, which one aren't you understanding?

    Line 2. I'm assuming he took the (k + 1)^2 from the one just after k^2.
    But the (k+1)^3 is gone and now there is 4(k+1) which I don't understand
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    (Original post by NinCheng)
    Line 2. I'm assuming he took the (k + 1)^2 from the one just after k^2.
    But the (k+1)^3 is gone and now there is 4(k+1) which I don't understand
    If you had something like \displaystyle x^3 + x^2 you would factor out an x^2 to get \displaystyle x^2(x + 1)

    Now if you had \displaystyle 2x^3 + x you could factor out an x to get \displaystyle x(2x^2 + 1) alternatively though you could factor out 2x and get \displaystyle 2x(x^2 + \frac{1}{2})

    So when he has \displaystyle \frac{1}{4}k^2(k+1)^2 + (k+1)^3 he is factoring out \displaystyle \frac{1}{4}(k+1)^2 from both terms.

    He does this because the required form needs a \displaystyle \frac{1}{4}(k+1)^2 at the front, so you may aswell factor it out early and make it easy
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    (Original post by DylanJ42)
    If you had something like \displaystyle x^3 + x^2 you would factor out an x^2 to get \displaystyle x^2(x + 1)

    Now if you had \displaystyle 2x^3 + x you could factor out an x to get \displaystyle x(2x^2 + 1) alternatively though you could factor out 2x and get \displaystyle 2x(x^2 + \frac{1}{2})

    So when he has \displaystyle \frac{1}{4}k^2(k+1)^2 + (k+1)^3 he is factoring out \displaystyle \frac{1}{4}(k+1)^2 from both terms.

    He does this because the required form needs a \displaystyle \frac{1}{4}(k+1)^2 at the front, so you may aswell factor it out early and make it easy
    Oh, that makes sense.

    Thanks!
 
 
 
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