1 / F(1)*F(3) + 1 / F(3)*F(5) + ... + 1 / F(2*n+1)*F(2*n+3) ---> phi^(-1) ? (Fibonacci #)

Fibonacci - so many formulas, so few time ...

Given that its not a Mathcad questions and probably also can't be done/proven with Mathcad, I guess chances are rather low that anyone will present a formal proof here.

Maybe you find here

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormulae.html

and/or in the reference section at the end something of interest.

Maybe you are especially interested at Vajda

https://www.amazon.com/exec/obidos/ASIN/0486462765/fibonacnumbersan

or Johnson

www.maths.dur.ac.uk/~dma0rcj/PED/fib.pdf