Question

The Harmonic mean of 3, 7, 8, 10, 14 is:
(a) \[\dfrac{3+7+8+10+14}{5}\]
(b) \[\dfrac{5}{3+7+8+10+14}\]
(c) \[\dfrac{\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{14}}{5}\]
(d) \[\dfrac{5}{\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{14}}\]

Answer 1

Hint: In order to solve this question, we should know that harmonic progression are those whose terms when reciprocated, they form an arithmetic progression, that means if a, b, c are in a harmonic progression, then \[\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}\] should be in arithmetic progression.

Complete Step-by-Step solution:
In this question, we have to find the harmonic mean of 3, 7, 8, 10, and 14. We know that when the terms are in harmonic progression, then their reciprocals are in arithmetic progression. Therefore, we can say,
\[\dfrac{1}{3},\dfrac{1}{7},\dfrac{1}{8},\dfrac{1}{10},\dfrac{1}{14}\] are in arithmetic progression. Also, we know that the harmonic mean is the reciprocal of the arithmetic mean. So, if we will find the arithmetic mean of the arithmetic progression, then we can find the harmonic means. We also know that arithmetic mean of n number of terms is the average sum of those n terms. So, we can say that the arithmetic mean of \[\dfrac{1}{3},\dfrac{1}{7},\dfrac{1}{8},\dfrac{1}{10},\dfrac{1}{14}\] is \[\dfrac{1}{x}\], that is
\[\dfrac{1}{x}=\dfrac{\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{14}}{5}\]
Now, we can say that the harmonic mean will be reciprocal of \[\dfrac{1}{x}\], that is x and we know that
\[x=\dfrac{5}{\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{14}}\]
Hence, option (d) is the right answer

Note: In this question, we can also find the solution using the options because we know that the reciprocal of the harmonic mean is the arithmetic mean of the reciprocal of terms. So, option (d) is the correct answer. The common mistake that students can make is by not taking the reciprocal and choosing option (c) as the right answer.