Patterns give rise to problems

Algebra Level 3

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

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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