I have the following sequence \(\displaystyle (x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}\)

Also I know that \(\displaystyle a,b,c\in \mathbb{R}\) and \(\displaystyle |c|<1,\ b\neq 1, \ |bc|<1\)

I need to find the limit of \(\displaystyle x_{n}\).

My attempt is in the picture.The result should be \(\displaystyle \frac{ac}{(1-bc)(1-c)}\)

I miss something at these two sums which are geometric progressions.Each sum should start with \(\displaystyle 1\) but why ? If k starts from 0 results the first terms are \(\displaystyle bc\) and \(\displaystyle c\) right?