<s>[INST] What is the limit of $2c^{n-1}-c^{n-2}-...-c-1$ when $n\to\infty$ I have a sequence in which the value of the $n^{th}$, $x_n$ is given by
$$x_n=2c^{n-1}-c^{n-2}-...-c-1$$
Where $c>0,$
then in the limit when $n\to\infty$ what is $x_n$?
 [/INST] First simplify the expression. When $c \ne 1$, we have
$$\begin{align*}
2c^{n-1} - c^{n-2} - \cdots - c - 1
&= 2c^{n-1} - \sum_{i=0}^{n-2} c^i \\
&= 2c^{n-1} - \dfrac{c^{n-1} - 1}{c-1} \\
&= \frac{2c^n-3c^{n-1}+1}{c-1}
\end{align*}$$
Now consider what happens if $0<c<1$ and if $c>1$ separately.
When $c = 1$, the expression is equal to $2-(n-1) = 3-n$, so tends to $-\infty$.
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