Mathematics - Limits Question with Solution | TestHub

 $\underset{x\to 1}{\lim }\frac{x+x^2+x^3+\dots .+x^n-n}{x-1}=820 \left(\frac{0}{0}\right)$form

Applying L'Hospital rule

$\underset{x\to 1}{\lim }\frac{1+2x+3x^2+\dots ..+n{x}^{n-1}}{1}=820$

$\Rightarrow 1+2+3+\dots \dots +n=820$

$\Rightarrow \frac{n(n+1)}{2}=820$

$\Rightarrow n^2+n=1640$

$\Rightarrow n^2+n-1640=0$

$\Rightarrow n=40, -41$

Since $n$ is natural number $\Rightarrow n=40$.