Ugly Fibonacci Sums

Algebra Level 4

If kk is an integer that satisfies the identity (i=1nF2i1)2(2i=1nF2i12)=(i=12n2(1)iFi2)k\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 kk.

Details and Assumptions\text{Details and Assumptions}

F1=F2=1F_1=F_2=1, and Fn=Fn1+Fn2F_n=F_{n-1}+F_{n-2}.

[You should assume that n2 n \geq 2 for the summations to make sense.]

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