Q114 / 160 · WBJEE 2011 · Mathematics

The value of \(\lim _{x \rightarrow 1} \frac{x+x^2+\ldots+x^n-n}{x-1}\) is

  • A
    \(\mathrm{n}\)
  • B
    \(\frac{\mathrm{n}+1}{2}\)
  • Correct answer
    C
    \(\frac{\mathrm{n}(\mathrm{n}+1)}{2}\)
  • D
    \(\frac{\mathrm{n}(\mathrm{n}-1)}{2}\)

Solution

Hints: \(\operatorname{Lt}_{x \rightarrow 1} \frac{(x-1)+\left(x^2-1\right)+\left(x^3-1\right) \ldots \ldots \ldots \ldots .\left(x^4-1\right)}{x-1}\) \(=1+2+3 \ldots \ldots \ldots \ldots \ldots \ldots+n=\frac{\mathrm{n}(\mathrm{n}+1)}{2}\)

Hence, the correct answer is option: 'C'.

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