# Inverse Fibonacci

Number Theory Level 4

$\displaystyle \sum_{i=2}^{\infty} \frac {F_i}{10^{i}}$

Let $$a$$ be defined as the summation above.

where $$F_i$$ are the Fibonacci numbers satisfying the relation $$F_{i+1}=F_i+F_{i-1}$$ with $$F_1=0,F_2=1$$.

and $$\alpha,\beta$$ be the two (not necessarily distinct) solutions of the quadratic equation $$a^2x^2-2ax+1-a^2=0$$

Find the value of $$\large{|\alpha^2-\beta^2|}$$

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