But where's the remaining information?

Algebra Level 4

\[\large x_iy^2+2(-1)^{i+1}x_{i+1}y+x_{i+2}=0\]

For each \(i \in \mathbb Z\), \(1 \le i \le n-2\) and \(x_i \ne 0\), the quadratic equation in \(y\) above has exactly one root, then find the value of the following expression.

\[\large\displaystyle\sum_{n=2}^{51}\sum_{k=1}^{n-1}\dfrac{x_1x_{k+1}}{x_kx_2}\]

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