<s>[INST] Calculate $\sum_{i=1}^{n-1} i\alpha^{2i}$ I try to calculate $\sum_{i=1}^{n-1} i\alpha^{2i}$
I think one week but still have no ideas to this sigma,thank everyone
 [/INST] Call $S = \sum_{i=1}^{n-1} i \alpha^{2i}=\sum_{i=1}^{n-1} i (\alpha^2)^{i}$ and $\alpha^2 = \beta$. If $\beta \neq 1$, then \begin{align} \beta S = \sum_{i=1}^{n-1} i \beta^{i+1} &= \sum_{i=1}^{n-1}\left((i+1)-1\right) \beta^{i+1} \\ &=\sum_{i=1}^{n-1} (i+1)\beta^{i+1} - \sum_{i=1}^{n-1} \beta^{i+1} \\ &= \sum_{i=2}^{n}i \beta^{i} - \sum_{i=2}^{n}\beta^{i} \\ &= \left( n\beta^n-\beta \right) +\sum_{i=1}^{n-1}i\beta^{i} - \sum_{i=2}^{n}\beta^{i} \\ &= \left( n\beta^n-\beta \right) + S - \frac{\beta^{n+1} - \beta^2 }{\beta  -1 }
 \end{align}
Thus $$ (\beta - 1)S = n\beta^{n}-\beta - \frac{\beta^{n+1} - \beta^2}{\beta -1 } $$
or \begin{align} S &= \frac{n\beta^n -\beta}{\beta -1} - \frac{\beta^{n+1} - \beta^2 }{(\beta - 1)^2
} \\ &=\frac{(n-1)\beta^{n+1} - n\beta^n + \beta}{ (\beta - 1)^2} \\& = \frac{(n-1)\alpha^{2n+2} - n\alpha^{2n} + \alpha^2}{ (\alpha^2 - 1)^2} 
\end{align}
If $\beta = \alpha^2 = 1$, then $S=\sum_{i=1}^{n-1} i (\alpha^2)^{i} = \sum_{i=1}^{n-1}i=\frac{(n-1)n}{2}$.
</s>