Let $[n]\_q=1+q+\cdots+q^{n-1}$. Is there a nice closed form of $\sum\_{s=1}^i[s]\_{q}$? One would expect that the answer will be some q-analog of $\frac{i(i+1)}{2}$, since $\sum\_{s=1}^i s=\frac{i(i+1)}{2}$.


Also I'm quite unfamiliar with q-theory, so if my terminology/notation is imprecise please let me know!



Does this q-analogue have a nice closed form?
$[n]\_q=\frac{q^n-1}{q-1}$


$\Sigma\_{s=1}^i\frac{q^s-1}{q-1}=\frac{(q-1)+(q^2-1)+...+(q^i-1)}{q-1}=\frac{1+q+q^2+...+q^i-(i+1)}{q-1}=\frac{q^{i+1}-1}{(q-1)^2}-\frac{i+1}{q-1}=\frac{[i+1]\_q-(i+1)}{q-1}$


