Good luck!Proving the Relationship between Fibonacci and Lucas Series

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The discussion focuses on proving the relationship between the Fibonacci and Lucas series, specifically the identity a2n = an * bn. It introduces the Lucas series formula bn = an-1 + an+1 for n ≥ 2 and highlights key identities involving the Golden Ratio. Participants suggest using substitution and induction to simplify the proof. The conversation encourages sharing progress for more effective assistance. The relationship between these two series is mathematically significant and can be explored through established formulas.
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Fibonacci and lucas series...

Let a1,a2,a3...,an be the numbers of fibonacci series...
Let b1,b2...bn be the number of lucas series.

bn=an-1 + an+1 for n\geq2

T.P.T : a2n=an*bn
 
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Suk-Sci said:
T.P.T : a2n=an*bn

Keep in mind the following two identities...

(Lucas_(n-1) + Lucas_(n+1))/5 = Fibonacci_n
Lucas_n = (Golden Ratio)^n + (-1)^n(Golden Ratio)^-n

... where the Golden Ratio = ((sqrt 5) + 1)/2
 


Hi, Suk-Sci,
you can substitute the given expression for the b's into the equation you want to prove; then you will have something only in terms of a's, that you can prove using induction.

If you want more help, try to show what you have done so far; that helps us help you. :)
 

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