Question: Problem with solving system

How can I solve the following system in Maple for $S_1$ and $S_{i+1}$? I have the code written, but it is giving me nothing as output.

eq1 := Q-A*S[1]*C/X+B*D*(sum(S[j], j = 2 .. i+1))/Y-r[1]*S[1] = 0;
eq2 := A*S[i-1]*C/X-A*S[i]*C/X-B*S[i]*D/Y-r[1]*S[i] = 0;
eq3 := A*S[i]*C/X-B*S[i+1]*D/(Y+S[i+1])-r[2]*S[i+1] = 0;
solve({eq1, eq2, eq3}, {S[1], S[i+1]});

The only non-constants in the system are the $S_j$'s for $j = 1, \ldots, i+1$.

Here is the system in math mode:
$$Q-\frac{AC}{X}S_1-r_1 S_1 +\frac{BD}{Y}\sum\limits_{j=2}^{i+1}S_j = 0 \\
\frac{AC}{X}S_{i-1} - \frac{AC}{X}S_i - \frac{BD}{Y}S_i - r_1 S_i = 0\\
\frac{AC}{X}S_i - \frac{BD S_{i+1}}{Y+S_{i+1}} - r_2 S_{i+1} = 0 $$

I have a head start on it by hand, but it's too cumbersome to complete.

Basically, my strategy is to solve for $S_{i+1}$ as a function of $S_1$ and some constants. Then to plug in $S_2, \ldots, S_{i+1}$ in the summation and solve for $S_1$ as a function of constants. Then I would obtain $S_{i+1}$ as a function of constants itself.

Thanks for any help.