What is the ratio of the number of ways to represent a number $a$ with a sum of $k$ numbers that are either $1$ or $2$, and $a+1$ with a sum of $k+1$ numbers that are either $1$ or $2$?
If there are $n$ ways to represent $a$, we can either put a $1$ in one of those $k$ places or change a $2$ to $1$ and put one more $1$ in one of those $k$ places. Is there a way to calculate this ratio without knowing the number of $2$'s in the representation of $a$?
I presume you are comparing $\binom{k}{a-k}$ with $\binom{k+1}{a-k}$? The ratio is $$\frac{k+1}{2k-a+1}.$$