Nimplesy wrote: Find all real number $x$ which could be represented as $x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$ Let us write $x=[1;a_1;...a_n]$ the given number. Obviously such real all are positive rational numbers. Note that $[1;a]\le 1$ Let $n\ge 2$ : If $[1;a_1;...;a_n]>1$, then $[1;a_1;...;a_{n-1}]$ $=a_n[1;a_1;...;a_n]-a_{n-1}$ $>a_n-a_{n-1}\ge 1$ And so $[1;a_1;...;a_n]>1$ $\implies$ $[1,a_1]>1$, impossible. So all these numbers are $\le 1$ Note then that $\frac 1q=[1,q]$ and that $\frac pq=[1;1;2;3;...;p-2;p-1;q]$ $\forall q\ge p>1$ Hence the answer : the required set is exactly the set of all rational numbers in the interval $(0,1]$