Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Anyone know how to explain this??

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Help.

(edited 6 years ago)

A picture would be helpful

OP

Original post by Y11_Maths

A picture would be helpful

there.

Study Forum Helper

Original post by LoveLifeAndPie

there.

What are

F_k

? Fibbonaci numbers or what??

OP

Original post by RDKGames

What are

F_k

? Fibbonaci numbers or what??

yeah.

I can’t see any pictures???

OP

Original post by Y11_Maths

I can’t see any pictures???

the link.

Study Forum Helper

\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.

(edited 6 years ago)

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