<s>[INST] Next Term Of Strange Sequence I tutored a 10th grader and I was asked this puzzle and I had spent nearly an hour with it and got “no where”. Any one can crack it? Please let me know. Thank you.
Question: Find the $14$ th term of the sequence: $$ \frac{1}{2},  \frac{3}{7}, \frac{1}{3}, \frac{5}{19}, \frac{3}{14}, .... $$.
 [/INST] $\displaystyle\frac24\ ,\ \frac37\ ,\ \frac4{12}\ ,\ \frac5{19}\ ,\ \frac6{28}\ ,...$ 
differences between denominators are $3,5,7,9,...$
(and numerators are consecutive integers $2,3,4,5,6,...$ )
$\displaystyle\frac24,\frac37,\frac4{12},\frac5{19},\frac6{28}, \\
\displaystyle\frac7{39},\frac8{52},\frac9{67},\frac{10}{84},\frac{11}{103},\\
\displaystyle\frac{12}{124},\frac{13}{147},\frac{14}{172},\frac{15}{199},\frac{16}{128}$ 
elementary watson, $\displaystyle a_{14}=\frac{15}{199}$ . 
More generally, $\displaystyle a_n = \frac{n+1}{n^2+3}$ .  
Using the formula, $\displaystyle a_{14}=\frac{14+1}{14^2+3}=\frac{14+1}{196+3}=\frac{15}{199}$ . 
</s>