If $\lim_{x \to 1} \frac{x + x^{2} + x^{3} + . . . + x^{n} - n}{x - 1} = 5050$ then n equal

(a) 10
(b) 100
(c) 150
(d) none of these

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Solution

(b) 100

$\lim_{x \to 1} \frac{x + x^{2} + x^{3} . . . x^{n} - n}{x - 1} = 5050 \Rightarrow \lim_{x \to 1} \frac{(x - 1)}{x - 1} + \frac{(x^{2} - 1)}{x - 1} + \frac{(x^{3} - 1)}{x - 1} . . . \frac{x^{n} - 1}{x - 2} = 5050 \Rightarrow 1 + 2 + 3 . . . n = 5050 [∵ \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}] \Rightarrow \frac{n (n + 1)}{2} = 5050 \Rightarrow n (n + 1) = 10100 \Rightarrow n (n + 1) = 100 \times 101 On c omparing : n = 100$

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