JEE Advance - Mathematics (1999 - No. 24)
For a positive integer $$n$$, let
$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$$. Then
$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$$. Then
$$a\left( {100} \right) \le 100$$
$$a\left( {100} \right) > 100$$
$$a\left( {200} \right) \le 100$$
$$a\left( {200} \right) > 100$$