Queston

Take any two real number R1, R2 . Buildup a series as follows R1, R2, R1+R2, R1+R2+R2, (R1+R2+R2)+(R1+R2),,,,,,Rn+1=Rn+Rn-1........ The same way we build the ordinary fibo ordinary series 1,1,2,3,5,8,13.....

As the series propagates the following ratio Rn+1/Rn will approach fibo(1.6180339....), which can be checked approximately experimentally by using finite real numbers. Is that possible to prove it based on the regular Fibo. series and also for irrational numbers: R1 and R2 ?

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$