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.