How can I verify the ratio of Lucas numbers to fibonacci numbers algebreically

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The article:

"The Lucas numbers 1,3,4...are the sums of alternate Fibonacci numbers. The ratios of Lucas to Fibonacci must satisfy: $R_j = \frac{F_{i+1}+F_{i-1}}{F_i}=\frac{2F_{i+1}}{F_i-1}$

I know that Fibonacci numbers and Lucas are the same except their starting values.

Can someone show how the ratio between the two equal the given condition that was set for $R_j$