A) 10
B) 102
C) 103
D) 104
Correct Answer: C
Solution :
It is given that\[k=\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{\sum\limits_{k=1}^{1000}{{{(x+k)}^{m}}}}{{{x}^{m}}+{{10}^{1000}}} \right)\] Divide numerator and denominator by \[{{x}^{m}},\], we have\[k=\underset{x\to \infty }{\mathop{\lim }}\,.\left[ \frac{\sum\limits_{k=1}^{1000}{{{\left( 1+\frac{k}{x} \right)}^{m}}}}{1+\frac{{{10}^{1000}}}{{{x}^{m}}}} \right]\] \[=\frac{1+1+1+.....\text{upto}\,\text{1000}\,\text{times}}{1+0}\]\[=1000={{10}^{3}}\] So, correct option is [c].
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