**Ans. (40.00)**

**Sol.** $\lim\limits_{x\to1}{x+x^2+x^3+.....+x^n-n\over x-1}=820\left({0\over0}\right)$

$\lim\limits_{x\to1}{x+2x+3x^2+.....+nx^{n-1}\over1}=820$

$\implies1+2+3+.....+n=820$

$\implies{n(n+1)\over2}=820$

$\implies n^2+n=1640$

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

$\implies n=40\quad\quad n\in N$