Question

What is the harmonic mean of the numbers $4,8,16$?
A. $6.4$
B. $6.7$
C. $6.85$
D. $7.8$

Answer 1

Hint: If the difference between two successive numbers given in a sequence of some numbers are equal then we say that the numbers are in arithmetic progression or simply, A.P. If three numbers are in arithmetic progression then we say that the reciprocals of the three numbers are in harmonic progression and always there exists a harmonic mean.

Formula Used:
Harmonic mean of three numbers $a,b,c$ is given by $\dfrac{3}{{\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}}}$ or $\dfrac{{3abc}}{{ab + bc + ca}}$

Complete step by step solution:
The given numbers are $4,8,16$
Here $a = 4,b = 8,c = 16$
Substituting these values in the formula, we get
Harmonic mean $ = \dfrac{{3 \times 4 \times 8 \times 16}}{{4 \times 8 + 8 \times 16 + 16 \times 4}}$
$\begin{array}{l} = \dfrac{{1536}}{{32 + 128 + 64}}\\ = \dfrac{{1536}}{{224}}\end{array}$
HCF of $1536$ and $224$ is $32$
Dividing 1536 by $32$, we get $48$
Dividing $224$ by $32$, we get $7$
So, the harmonic mean $ = \dfrac{{48}}{7} = 6.85$

Option ‘C’ is correct

Note: You need to clear concept about arithmetic progression and harmonic progression and the relation between them. Whenever some terms or numbers are given as arithmetic sequence, you need to understand that the reciprocals of the terms or the numbers are in harmonic progression. It is always true.
Steps to calculate HP
Step 1: Calculate the reciprocal (1/value) for every value.
Step 2: Find the average of those reciprocals, by just adding them and divide by the number of total values
Step 3: Then do the reciprocal of that average.