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So I was given the triangular array of numbers below (the first line consists of two "1")

$$11$$

$$1\frac{3}{2}1$$

$$1\frac{6}{4}\frac{6}{4}1$$

$$1\frac{10}{7}\frac{10}{6}\frac{10}{7}1$$

$$1\frac{15}{11}\frac{15}{9}\frac{15}{9}\frac{15}{11}1$$

and I was told to find a function of two variables $f(r,c)$ that takes as input the row number $r$ (starting with $1$) and the column number $c$ (starting with $0$) and outputs the correct number. So for example $f(3,1)=\frac{6}{4}$. The general statement is

$$f(r,c)=\frac{r(r+1)}{(r-c)(r-c+1)+c(c+1)}$$

My question is, how can I figure out if this triangle of numbers contains all rational numbers somewhere and if not, how do I figure out which fractions will never appear in this pattern?

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    None of the numbers will be less than $1$ and greater than $2$. Do you mean it will hit all rationals in $[1,2)$?2011-12-11

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