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Calculate the determinant

$$\begin{vmatrix} n & s_1 & s_2 & \cdots & s_{n-1} \\ s_1 & s_2 & s_3 & \cdots & s_n \\ s_2 & s_3 & s_4 & \cdots & s_{n+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots_\strut\\ s_{n-1} & s_n & s_{n+1} & \cdots & s_{2n-2} \end{vmatrix}$$

where $s_k=x_1^k+x_2^k+\cdots+x_n^k$.

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Are your $x_i$ real numbers? Also, have you computed this for a few values of $n$? Perhaps that might help answer your question in general. –  Prahlad Vaidyanathan Sep 1 '13 at 14:05
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1 Answer

HINT:

Let $x_i,1\le i\le n$ are the roots of $\sum_{1\le i\le n}a_ix^i=0$

So, if we set the first column $$C_1'=\frac{\sum_{1\le j\le n}a_jC_j}{a_1}$$

Using Newton's Sums,

$ a_1C_1'=\sum_{1\le j\le n}a_jC_j=\sum_{1\le j\le n}x^{j}(\sum_{1\le i\le n}a_ix^i)=\sum_{1\le j\le n}x^j\cdot0=0$

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