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Let FnF_nFn be the nnnth Fibonacci number, where F1=1F_1=1F1=1, F2=1F_2=1F2=1 and Fn+1=Fn+Fn−1F_{n+1}=F_n+F_{n-1}Fn+1=Fn+Fn−1, for n≥2n \geq 2n≥2. Evaluate the sum ∑n=2∞FnFn+1Fn−1.\sum_{n=2}^\infty \frac{F_n}{F_{n+1} F_{n-1}}.n=2∑∞Fn+1Fn−1Fn.
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