Ugly Fibonacci Sums

Algebra Level 4

If \(k\) is an integer that satisfies the identity \[\left(\sum_{i=1}^{n}F_{2i-1}\right)^2-\left(2\sum_{i=1}^{n}F_{2i-1}^2\right)=\left(\sum_{i=1}^{2n-2}(-1)^iF_i^2\right)-k\] then find \(k\).

\(\text{Details and Assumptions}\)

\(F_1=F_2=1\), and \(F_n=F_{n-1}+F_{n-2}\).

[You should assume that \( n \geq 2 \) for the summations to make sense.]

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