I am practicing finding patterns and sequences with numbers and found one that I cannot seem to find the logic. $$ \frac{2}{3} ,\frac{4}{3} ,\frac{4}{5} ,\frac{6}{5} ,\frac{6}{7},\frac{6}{8}... $$ The last term 6/8 breaks my understanding of a possible pattern. If the last term didn't exist, I could suggest the following. $$ \frac{2}{3} ,\frac{4}{3} ,\frac{4}{5} ,\frac{6}{5} ,\frac{6}{7},\frac{8}{7},\frac{8}{9},\frac{10}{9}... $$
Is it possible that the original sequence does not have a pattern with which I can find the nth term?
For $$\{2,4,4,6,6,6\}$$ there are $27$ sequences in $OEIS$ and for $$\{3,3,5,5,7,8\}$$ there are $30$ other.
Now, for $$\frac{2}{3} ,\frac{4}{3} ,\frac{4}{5} ,\frac{6}{5} ,\frac{6}{7},\frac{6}{8}...$$ it is hard to say