Question

The value of $\underset{x\to 1}{lim}\frac{x+x^2+...+x^n-n}{x-1}$ is

• A. $n$
• B. $\frac{n+1}{2}$
• C. $\frac{n(n+1)}{2}$
• D. $\frac{n(n-1)}{2}$

Answer 1

The correct option is C $\frac{n(n+1)}{2}$

$\underset{x\to 1}{lim}\frac{x+x^2+...+x^n-n}{x-1}$
$=\underset{x\to }{lim}\frac{(x-1)(x^2-1)+......(x^n-1)}{(x-1)}$
$=\underset{x\to 1}{lim}1+(x+1)+(x^2+x+1)+......+n$ terms
$=1+2+3+.....+n$

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