Odd-sized groups of geometric terms

The sum of the first $n$ terms, $S_n$, is $S_n=\frac1a[1-(a-1)^n]$, where $a$ is a constant, $a\ne1$, $n\in\mathbb Z^+$. Obtain the $n$th term $T_n$ and prove that “$S_n$ is a geometric series” (printed wording).3 marks

The sequence $\{T_n\}$ is grouped as $(T_1),(T_2,T_3,T_4),(T_5,T_6,T_7,T_8,T_9),\ldots$, each subsequent bracket having 2 more terms than the previous. Find:

(i) total terms in the first $n$ brackets 2 marks;

(ii) the middle term of bracket 11 in terms of $a$ 2 marks;

(iii) the range of $a$ for the sum to infinity to exist; and the least $n$ for the sum of all terms in the first $n$ brackets to be within 0.1% of the sum to infinity when $a=\frac{39}{20}$.4 marks