nafiul9090 wrote:A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.
For example, the Farey sequence of order 3 is: {0, 1/3, 1/2 , 2/3 , 1}.
Is sequence S a Farey sequence?

(1) Sequence S has fewer than 10 elements.
(2) The second element of sequence S is 1/5 .

Important:

1. "Fractions between 0 and 1" must (and will) be considered as "fractions between 0 and 1, both included" so that the example given satisfies the definition presented.
2. The { } notation will always denote here a finite sequence. (In other words, the order of the elements presented is relevant.)
$$S\,\,\mathop = \limits^? \,\,{\rm{Farey}}$$
$$\left( 1 \right)\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,S = \left\{ {0;{1 \over 3};{1 \over 2};{2 \over 3};1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {{\rm{given}}} \right)\,\, \hfill \cr
\,{\rm{Take}}\,\,S = \left\{ {0;1;{1 \over 3};{1 \over 2};{2 \over 3}} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\left( {{\rm{wrong}}\,\,{\rm{order}}} \right)\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,S = \left\{ {0;{1 \over 5};{2 \over 5};{3 \over 5};{4 \over 5};1;{1 \over 4};{2 \over 4} = {1 \over 2};{3 \over 4};{1 \over 3};{2 \over 3}} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\left( {{\rm{wrong}}\,\,{\rm{order, }}\,{\rm{although}}\,\,{1 \over 5}\,\,{\rm{IS}}\,\,{\rm{the}}\,{\rm{ second}}\,\,{\rm{here!}}} \right)\,\, \hfill \cr
\,{\rm{Take}}\,\,S = \left\{ {0;{1 \over 5};{2 \over 5};{3 \over 5};{4 \over 5};1;{1 \over 4};{2 \over 4} = {1 \over 2};{3 \over 4};{1 \over 3};{2 \over 3}} \right\}\,\,{\rm{but}}\,\,{\rm{rewritten}}\,\,\,{\rm{in}}\,\,{\rm{increasing}}\,\,\,{\rm{order}}\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\left( {{1 \over 5}\,\,{\rm{WILL}}\,\,{\rm{BE}}\,\,{\rm{the}}\,\,{\rm{second!}}} \right)\,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\left( {{1 \over 5} \in \,\,S\,\,{\rm{Farey}}\,\,\,\, \Rightarrow \,\,\,{\rm{S}}\,\,{\rm{has}}\,\,{\rm{more}}\,\,{\rm{than}}\,\,{\rm{10}}\,\,{\rm{elements}}!} \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: In approximately 2012 (when I was very active in this forum), I made a really good friendship with the great Anurag (in exchanging comments from posts).
He was an extraordinary expert who could correct us, or be corrected by us (experts or not), without taking his ego into the matter.
If anyone who still has contact with him read this post, please send to him my very best wishes. Thank you!