Answer

Verified

357.9k+ views

**Hint:**The reciprocal of first n natural numbers is given by

$ \Rightarrow $Reciprocal of first n natural numbers$ = 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},........,\dfrac{1}{n}$

Now, the harmonic mean is the reciprocal of the arithmetic mean. The Harmonic Mean can be given by the formula

$ \Rightarrow $Harmonic Mean (H.M.)$ = \dfrac{n}{{\sum\limits_{i = 1}^n i }}$

Using this formula, we will get our answer.

**Complete step by step solution:**

In this question, we are asked to find the Harmonic Mean (H.M.) of the reciprocal of first n natural numbers.

By definition harmonic mean is the reciprocal of arithmetic mean.

Suppose we are given some data ${X_1},{X_2},{X_3},.......,{X_n}$ then the harmonic mean of this ungrouped data is given by

$ \Rightarrow $Harmonic Mean (H.M.)$ = \dfrac{n}{{\sum\limits_{n = 1}^n {{X_n}} }}$

So, here we have to find the Harmonic Mean of the reciprocal of first n natural numbers. Now, first n natural numbers are given by

$ \Rightarrow $First n natural numbers$ = 1,2,3,.......,n$

And the reciprocal of first n natural numbers is given by

$ \Rightarrow $Reciprocal of first n natural numbers$ = 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},........,\dfrac{1}{n}$

Now, the sum of $1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},........,\dfrac{1}{n}$ is given by

$1 + \dfrac{1}{2} + \dfrac{1}{3} + ..... + \dfrac{1}{n} = \dfrac{{n\left( {n + 1} \right)}}{2}$

Therefore, we get

$ \Rightarrow $Harmonic Mean (H.M.)$ = \dfrac{n}{{\sum\limits_{i = 1}^n {\left( {1 + \dfrac{1}{2} + \dfrac{1}{3} + ...... + \dfrac{1}{n}} \right)} }}$

$ \Rightarrow $Harmonic Mean (H.M.)$ = \dfrac{n}{{\dfrac{{n\left( {n + 1} \right)}}{2}}}$

$ \Rightarrow $Harmonic Mean (H.M.)$ = \dfrac{2}{{n + 1}}$

**Hence, the Harmonic Mean of the reciprocal of first n natural numbers is $\dfrac{2}{{n + 1}}$. So, the correct option is option (C).**

**Note:**

> While finding the harmonic mean of all constant numbers (c), the H.M. will be also c.

> As compared to arithmetic mean and geometric mean, the Harmonic mean has the least value, that is

$\text{AM > GM > HM}$

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Write a letter to the principal requesting him to grant class 10 english CBSE