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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!

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$[n]_q=\frac{q^n-1}{q-1}$

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

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