5

Consider a geometric sequence with first term $1$ and common ratio $10$.
Let $S_n$ be the sum of the first $n$ terms of the sequence.

(a) Find an expression for $S_n$ in the form $\dfrac{a^{\,n}-1}{b}$, where $a,b\in\mathbb Z^{+}$.

(b) Hence, show that
$S_1+S_2+S_3+\dots+S_n=\frac{10\bigl(10^{\,n}-1\bigr)-9n}{81}.$