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

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

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!