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    A picture would be helpful
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    (Original post by Y11_Maths)
    A picture would be helpful
    there.
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    (Original post by LoveLifeAndPie)
    there.
    What are F_k? Fibbonaci numbers or what??
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    (Original post by RDKGames)
    What are F_k? Fibbonaci numbers or what??
    yeah.
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    (Original post by Y11_Maths)
    I can’t see any pictures???
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    \begin{aligned} s(x) & = \sum_{k=0}^{\infty} F_kx^k \\ & = F_0+F_1x + \sum_{k=2}^{\infty} F_{k}x^k \\ & = 0+x + \sum_{k=2}^{\infty} (F_{k-1}+F_{k-2})x^k \\ & = x+\sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\ & = x+\sum_{a=1}^{\infty} F_{a} x^{a+1} + \sum_{b=0}^{\infty} F_{b} x^{b+2} \\ & = x+x\sum_{a=1}^{\infty} F_a x^a + x^2 \sum_{b=0}^{\infty} F_b x^b  \\ & = x+x\sum_{a=0}^{\infty} F_a x^a + x^2 \sum_{b=0}^{\infty} F_b x^b \end{aligned}

    \begin{aligned} \qquad & = x+x \sum_{k=0}^{\infty} F_k x^k + x^2 \sum_{k=0}^{\infty} F_k x^k \\ & = x+xs(x)+x^2s(x) \end{aligned}


    Second line - all you do here is remove the first two terms out of sigma.

    Third line - all you do here is replace F_0=0 and F_1=1 since these are the first two Fibonnaci numbers.

    Fourth line - all you do is split the sum.

    Fifth line - make a change of variable for the first sum, which is a=k-1, and a change of variable for the second sum b=k-2

    Sixth line - x^{a+1}=x\cdot x^a so you can factor the one x out of the sum. Similarly for the second sum.

    Seventh line - note that F_0=0 so starting that sum from a=1 is the same as starting the sum from a=0, as then you'd just be adding on a 0 and this doesn't change the value.

    Eighth line - a,b are dummy variables and can be replace with k again.

    Last line - just apply the equality you start with and replace the sums.
 
 
 
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