MHB Limit of $(x_{n})_{n\geq 1} with Given Conditions

[Vali]

Hi!

I have the following sequence $$(x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}$$
Also I know that $a,b,c\in \mathbb{R}$ and $|c|<1,\ b\neq 1, \ |bc|<1$
I need to find the limit of $x_{n}$.

My attempt is in the picture.The result should be $\frac{ac}{(1-bc)(1-c)}$
I miss something at these two sums which are geometric progressions.Each sum should start with $1$ but why ? If k starts from 0 results the first terms are $bc$ and $c$ right?
View attachment 8779

 

[MountEvariste]

I believe the correct set up is $\displaystyle x_n = \sum_{j=1}^{n+1}\bigg(\sum_{k=0}^{j-1}ab^k\bigg)c^j = \sum_{j=1}^{n+1}\frac{a(b^j-1)}{b-1}c^j$ so that $\displaystyle \lim_{ n \to \infty} x_n = \sum_{j=1}^{\infty}\frac{a(b^j-1)}{b-1}c^j = \frac{ac}{(c-1)(bc-1)}.$

 

[Vali]

I'm stupid, I got the correct answer.I just needed to solve some little calculations.I don't know I thought I'm wrong..
Thanks!

 

[MountEvariste]

I'm stupid, I got the correct answer.I just needed to solve some little calculations.I don't know I thought I'm wrong..
Thanks!

I see. I didn't go through your calculations because I couldn't zoom in tbh.