# 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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