calculus - Find an explicit formula $1+(1+x)+(1+x+x^2)+cdots$

Find the sum of $n$ terms of the following series:
$$1+(1+x)+(1+x+x^2)+\cdots$$

The $n^{th}$ term $(t_n)$ is $\displaystyle\frac{x^n-1}{x-1}$, since each term is a Geometric Progression with common ratio $x$.

Now, I want to find $\displaystyle\sum_{n=1}^nt_n$. Is it possible to get a telescoping series here?

Answer

HINT:

if $x\ne1,$

$$\sum_{1\le r\le n }\frac{x^r-1}{x-1}=\frac{\left(\sum_{1\le r\le n }x^r\right)-n}{x-1}$$

Observe the Geometric Progression in the numerator