Question

The reciprocal of the arithmetic mean of the reciprocals,
A) ${\rm{AM}}$
B) ${\rm{GM}}$
C) ${\rm{HM}}$
D) None of these

Answer 1

Hint: Consider, arithmetic mean formula ${\rm{AM}} = \dfrac{{b + a}}{2}$. Now take the reciprocal of the terms ‘a’ and ‘b’ and substitute back in the formula. After solving you will arrive at the solution.

Complete step-by-step answer :
We know that, the formula for arithmetic mean for the sequence, $a$, ${\rm{AM}}$, $b$ is
$A.M = \dfrac{{b + a}}{2}$
Now for the sequence $\dfrac{1}{a}$, ${\rm{AM}}$, $\dfrac{1}{b}$,
The arithmetic mean becomes,
$\Rightarrow {\rm{AM = }}\dfrac{{\dfrac{1}{b} + \dfrac{1}{a}}}{2}$
Taking LCM and solving we get,
$\Rightarrow {\rm{AM = }}\dfrac{{a + b}}{{2ab}}$
Taking reciprocal on both the sides,
$\Rightarrow \dfrac{1}{{{\rm{AM}}}}{\rm{ = }}\dfrac{{2ab}}{{a + b}}$---(1)
We know that, harmonic mean is given by ${\rm{HM = }}\dfrac{{2ab}}{{a + b}}$---(2)
On comparing equation (1) and equation (2)
i.e., $\dfrac{1}{{{\rm{AM}}}} = {\rm{HM}}$
Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.
Hence, option (C) is correct.

Note: To solve this equation you need to know the concept of arithmetic mean and harmonic mean. Also, it is important to know that harmonic mean is reciprocal of arithmetic mean which will help you in solving other problems.