Patterns give rise to problems

Algebra Level 4

n=110010n{2n+2n}\large\sum_{n=1}^{100} 10^n\{2^n+2^{-n}\}

If the above summation can be evaluated to be abac\dfrac{a^b-a}{c} where a,ba,b and cc are positive integers and a,ca,c are coprime, find the value of a+b+ca+b+c.

Notation: {} \{ \cdot \} denotes the fractional part function.

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