Q. For a positive integer $n$, Let $a(n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{\left( 2^{n} \right) - 1}.$ Then
A) $a(100) \leq 100$
B) $a(100) > 100$
C) $a(200) \leq 100$
D) None of these
Explanation
(a)
It can be proved with the help of mathematical induction that $\frac{n}{2} < a(n) \leq n$.
$\therefore\frac{200}{2} < a(200) \Rightarrow a(200) > 100$
and $a(100) \leq 100$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.