Patterns give rise to problems

Algebra Level 4

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

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

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

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