Question

If ${lim}_{x\to 1}\frac{x+x^2+x^3\cdots x^n-n}{x-1}=5050,$ then n equal

• A. 10
• B. 100
• C. 150
• D. None of these

Answer 1

The correct option is B

100

${lim}_{x\to 1}\frac{x+x^2+x^3\cdots x^n-n}{x-1}=5050$
${lim}_{x\to 1}\frac{(x-1)}{x}-1+\frac{(x^2-1)}{x-1}+\frac{(x^3-1)}{x}-1\cdots \frac{x^n-1}{x-2}=5050$
$\Rightarrow 1+2+3\cdots n=5050[\because \frac{x^n-a^n}{x-a}=n{a}^{n-1}]$
$\Rightarrow \frac{n(n+1)}{2}=5050$
$\Rightarrow n(n+1)=10100$
$\Rightarrow n(n+1)=100\times 101$
on comparing :
$n=100$