Question

If n HM's are inserted between a & b, Harmonic mean of sequence formed is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let $a_1$, $a_2$, ------- $a_n$ be the HM's inserted.

a, $a_1$, $a_2$, ------- $a_n$, b form H.P.

$\Rightarrow$ $\frac{1}{a}$, $\frac{1}{a_1}$, $\frac{1}{a_2}$,--------$\frac{1}{a_n}$, $\frac{1}{b}$ form A.P.

Harmonic mean = $\frac{(n+2)}{\frac{1}{a}+\frac{1}{a_1}+-------+\frac{1}{a_n}+\frac{1}{b}}$

= $\frac{(n+2)}{\frac{n+2}{2}[\frac{1}{a}+\frac{1}{b}]}$ [sum of n terms in AP with first and last terms as 'a' & 'l' = $\frac{n}{2}$(a + l)]

= $\frac{1}{\frac{1}{2}\left(\frac{a+b}{ab}\right)})$

= $\frac{2ab}{a+b}$