Q. If $a,a_{1},a_{2},\ldots,a_{2n},b$ are in arithmetic progression and $a,g_{1},g_{2},\ldots,g_{2n},b$ are in geometric progression and $h$ is the harmonic mean of $a$ and $b$, then $\frac{a_{1} + a_{2n}}{g_{1}g_{2n}} + \frac{a_{2} + a_{2n - 1}}{g_{2}g_{2n - 1}} + ... + \frac{a_{n} + a_{n + 1}}{g_{n}g_{n + 1}}$ is equal to
A) $2nh$
B) $\frac{n}{h}$
C) $nh$
D) $\frac{2n}{h}$
Explanation
(d)
By the properties of AP and GP
$a_{1} + a_{2n} = a_{2} + a_{2n - 1} = ... = a_{n} + a_{n + 1} = a + b$
and $g_{1}g_{2n} = g_{2}g_{2n - 1} = ... = g_{n}g_{n + 1} = ab$
$\therefore\ \frac{a_{1} + a_{2n}}{g_{1}g_{2n}} + \frac{a_{2} + a_{2n - 1}}{g_{2}g_{2n - 1}} + ... + \frac{a_{n} + a_{n + 1}}{g_{n}g_{n + 1}}$
$= \frac{a + b}{ab} + \frac{a + b}{ab} + ... + \frac{a + b}{ab}$
$= \frac{n(a + b)}{(ab)} = \frac{2n}{h}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.