Question

Two AMs $A_1$ and $A_2$, two GMs $G_1$ and $G_2$ and two HMs $H_1$ and $H_2$ are inserted between two numbers a and b, then

$\frac{1}{H_1}$ + $\frac{1}{H_2}$ equals.

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

If a,b,c,d are in A.P, a+d = b+c

If a,b,c,d are in G.P, ad =bc

a, $H_1$ , $H_2$, b are in H.P

$\Rightarrow$ $\frac{1}{a^{{}^{\prime }}}$,$\frac{1}{H^{{}^{\prime }}}$,$\frac{1}{H_2^{{}^{\prime }}}$,$\frac{1}{b}$ are in A.P

$\Rightarrow$ $\frac{1}{a}$ + $\frac{1}{a}$ = $\frac{1}{H_1}$ + $\frac{1}{H_2}$

$\Rightarrow$ $\frac{1}{H_1}$ + $\frac{1}{H_2}$ = $\frac{a+b}{ab}$-----------------(1)

a, $A_1$ , $A_2$, b are in A.P

$\Rightarrow$ a+b = $A_1$ + $A_2$

a, $G_1$ , $G_2$, b are in G.P

$\Rightarrow$ ab = $G_1$ $G_2$

Substituting in (1), $\frac{1}{H_1}$ + $\frac{1}{H_2}$ = $\frac{A_1+A_2}{G_1{G}_2}$