Question

If $a_1, a_2, a_3,...., a_n$ are $n$ distinct odd numbers not divisible by any prime greater than $5$, then $\frac{1}{a_{1}} +\frac{1}{a_{2}} +.....+\frac{1}{a_{n}} $

• A. $< 1$
• B. $= 1$
• C. $= 2$
• D. $< 2$

Answer 1

Answer: (D)

We observe that all the terms of $\frac{1}{a_{1}} +\frac{1}{a_{2}} +.....+\frac{1}{a_{n}}$ are contained in
$ \left(1+\frac{1}{3}+\frac{1}{3^{2}}+.....\right)\left(1+\frac{1}{5}+\frac{1}{5^{2}}+.....\right)$.
$\Rightarrow \frac{1}{a_{1}}+\frac{1}{a_{2}}+.....+\frac{1}{a_{n}} < \left(\frac{1}{1-\frac{1}{3}}\right)\left(\frac{1}{1-\frac{1}{5}}\right)$
$= \frac{3}{2}\cdot\frac{5}{4} <2$