I've been reading a book on number theory by Hardy and Wright and came across two properties of Farey Series: namely, if $\frac{h}{k},\frac{h'}{k'},\frac{h''}{k''}$ are sequential terms, then the following two theorems are true: $$hk'-h'k=1$$ $$\frac{h'}{k'}=\frac{h''+h}{k''+k}$$

I am attempting to prove that these two are equivalent. The first implies the second is easy to prove, but the other way is tougher. I know it can be done by induction over the Farey Series parameter but I want to attempt it without such. I've tried loads of methods and paths with no clear success. Any hints or advice?

  • $\begingroup$ Since the $(n+1)$th Farey series is defined recursively from the $n$th, I dk why you wouldn't want to prove it by induction on $n$, especially if you don't have a non-recursive or non-inductive def'n of $F_n.$ $\endgroup$ – DanielWainfleet May 31 '17 at 17:03
  • $\begingroup$ I actually didn't have a recursive definition. The definition of the nth Farey Series I have is the ordered sequence of all rationals between 0 and 1 inclusive with denominators no greater than n. $\endgroup$ – Vedvart1 Jun 1 '17 at 12:52

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