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Math Help - One confusing step in a solution

  1. #1
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    One confusing step in a solution

    Hi,

    I'm trying to understand the solution to a problem, and I get most of it, but there is one step in the algebra I can't quite follow. Please help!
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  2. #2
    Master Of Puppets
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    It looks like the \frac{1+\sqrt{5}}{2} and the \frac{1-\sqrt{5}}{2} terms have been grouped together, then factored.
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  3. #3
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    Hello, stevey!

    pockslides is correct . . .


    \displaystyle{\frac{1}{\sqrt{5}}\left(\frac{1+\sqr  t{5}}{2}\right)^{n-1}\!\!\! - \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1} \!\!\!+ \frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right  )^2 \!- \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n

    . . \displaystyle =\;\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\ri  ght)^{n-1}\!\!\left(\frac{1+\sqrt{5}}{2}+1\right) - \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\!\!\left(\frac{1-\sqrt{5}}{2} + 1\right)

    Watch very carefully . . .


    \displaystyle{\underbrace{\frac{1}{\sqrt{5}}\left(  \frac{1+\sqrt{5}}{2}\right)^{n-1}}_{A}\!\!\! - \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}}_{B} \!\!\!+ \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{  5}}{2}\right)^n}_{C} \!- \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n}_{D}


    . . \displaystyle =\; \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{  5}}{2}\right)^n}_{C}\!\!\! + \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}}_{A} \!\!\! - \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{  5}}{2}\right)^n}_{D} \!- \underbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n}_{B}


    . . \displaystyle =\; \overbrace{\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5  }}{2}\right)^{n-1}\!\!\left(\frac{1+\sqrt{5}}{2} + 1\right)}^{\text{Factor }C\text{ and }A} - \overbrace{\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\!\!\left(\frac{1-\sqrt{5}}{2} + 1\right)}^{\text{Factor }D\text{ and }B}
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